Fundamentals of financial derivatives nrp

FUNDAMENTALS OF

FINANCIAL DERIVATIVES

FUNDAMENTALS OF FINANCIAL DERIVATIVES

N.R. Parasuraman

SDM-IMD SDM Institute for Management Development

Mysore

PREFACE Few topics in Finance have undergone the type of change that Derivatives have over

the last few years. Dealers and Corporate practitioners have discovered several new

uses to Derivatives, resulting in lowering risk and optimizing return. Many students of

Financial Management and practitioners find the basic tenets of these Derivatives

difficult to understand in the beginning. Standard text books give answers to their

queries but since these are embedded in a cluster of applicable theory, exceptions and

mathematical notations, the beginner is often confounded.

The principal objective behind attempting this work is to help the newcomer to the world

of Derivatives get a grip on various facets in a simple manner. The attempt has been to

dwell on the most important characteristics of these instruments, without going into too

much of mathematical analysis. Let me hasten to add that in the process the work

cannot be a substitute for an advanced text book on the topic. What it seeks to

accomplish is to give the reader a quick and easy approach to understand the basic

complexities in day-to-day situations. Once comfortable with the basics, the reader is

advised to get a deeper understanding of various sub-topics with the help of text books

that cover the full mathematical application.

In completing this work, I am indebted to a number of people who encouraged me and

provided me with support. Dr. Jagadeesha, Professor and Chairman, DOS in

Management, KSOU was instrumental in convincing me that such a work would be useful

and in showing me the type of emphasis I should lay on various topics. Prof.

J.M.Subramanya, Director of SDM-IMD was very helpful at various stages of the work and

in generally reassuring me about the quality. Prof.Vinod Madhavan of SDM-IMD, helped

me by patiently going through the drafts and suggesting a number of changes.

Thanks are also due to my father N.P.Ramaswamy who was a source of inspiration and

encouragement and to my wife Prema, who spent long hours checking the drafts and

helping me in data entry. I also thank my colleagues at SDM-IMD for all their support. I

wish to make particular mention of the support of Mr.M.V.Sunil, Mr.M. Rangaswamy,

Ms.Madhura .S. Narayan and Ms.R. Gayithri in completing the final transcript of the book.

N.R.PARASURAMAN


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BRIEF CONTENTS Module 1 ............................................................................................ 8 FUTURES AND FORWARDS ......................................................... 8 1 Introduction to Derivatives Markets ......................................... 9 2 Forwards and Futures – a quick look...................................... 17 3 Hedging with Futures .............................................................. 28 4 Pricing of Futures and arbitrage conditions ........................... 41 5 Stock Index Futures ................................................................. 54 Module 2 – ....................................................................................... 69 INTRODUCTION TO OPTIONS ................................................... 69 1 Types of Options ....................................................................... 70 2 Pay off of various Options ........................................................ 79 3 Special applications of Options ................................................ 91 4 Options bounds- Calls............................................................. 104 5 Options bounds -Puts ............................................................. 113 Module 3 ........................................................................................ 121 ADVANCED TOPICS ON OPTIONS .......................................... 121 1 Option combinations............................................................... 122 2 Principles of Option Pricing – Put call parity. ...................... 141 3 The Binomial model for pricing of Options ........................... 153 4 The Black-Scholes model........................................................ 163 5 Volatility and Implied Volatility from the Black-Scholes model 172 Module 4 ........................................................................................ 180 OTHER DERIVATIVES AND RISK MANAGMENT ................. 180 1 Introduction to Options Greeks and Basic Delta Hedging ... 181 2 Interest Rate Derivatives and Eurodollar Derivatives......... 191 3 Swaps ...................................................................................... 203 4 Credit Derivatives................................................................... 216 5 Risk Management with Derivatives ...................................... 225 References ..................................................................................... 238


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DETAILED CONTENTS Module 1 ............................................................................................ 8 FUTURES AND FORWARDS ......................................................... 8 1 Introduction to Derivatives Markets ......................................... 9

1.1 Objectives ..............................................................................................9 1.2 Introduction...........................................................................................9 1.3 Derivatives – meaning and definition..................................................9 1.4 Types of Derivatives............................................................................10 1.5 Uses of Derivatives .............................................................................12 1.6 Derivatives in India ............................................................................14 1.7 Summary .............................................................................................15 1.8 Key words ............................................................................................15 1.9 Questions for Self- study.....................................................................15

2 Forwards and Futures – a quick look...................................... 17 2.1 Objectives ............................................................................................17 2.2 Introduction to Forwards and Futures ..............................................17 2.3 Basic hedging practices.......................................................................19 2.4 Cost of carry ........................................................................................23 2.5 Differences between Forwards and Futures ......................................25 2.6 Summary .............................................................................................26 2.7 Key words ............................................................................................27 2.8 Questions for Self- study.....................................................................27

3 Hedging with Futures .............................................................. 28 3.1 Objectives ............................................................................................28 3.2 Introduction.........................................................................................28 3.3 Long hedge and Short hedge ..............................................................29 3.4 Margin requirements for Futures ......................................................32 3.5 Basis risk .............................................................................................34 3.6 Cross hedging ......................................................................................37 3.7 Summary .............................................................................................39 3.8 Key words ............................................................................................40 3.9 Questions for Self- study.....................................................................40

4 Pricing of Futures and arbitrage conditions ........................... 41 4.1 Objectives ............................................................................................41 4.2 Introduction.........................................................................................41 4.3 Basic pricing principles.......................................................................42 4.4 Arbitrage opportunities ......................................................................44 4.5 Empirical evidence on cost of carry....................................................48 4.6 Rolling the hedge forward...................................................................49 4.7 Summary .............................................................................................51 4.8 Key words ............................................................................................52 4.9 Questions for Self- study.....................................................................52


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5 Stock Index Futures ................................................................. 54 5.1 Objectives ............................................................................................54 5.2 Introduction.........................................................................................54 5.3 Construction of stock indices ..............................................................55 5.4 Uses and applications of stock Index Futures ...................................56 5.5 Hedging with stock Futures ...............................................................57 5.6 Beta and the Optimal Hedge Ratio ....................................................61 5.7 Increasing and Decreasing Beta.........................................................63 5.8 Other uses of stock Futures................................................................64 5.9 Illustrations.........................................................................................66 5.10 Summary .............................................................................................67 5.11 Key words ............................................................................................68 5.12 Questions for Self- study.....................................................................68

Module 2 – ....................................................................................... 69 INTRODUCTION TO OPTIONS ................................................... 69 1 Types of Options ....................................................................... 70

1.1 Objectives ............................................................................................70 1.2 Introduction.........................................................................................70 1.3 Types of Options and option terminology ..........................................71 1.4 The question of exercise......................................................................74 1.5 Options markets..................................................................................75 1.6 Differences between Options and Futures.........................................76 1.7 Summary .............................................................................................77 1.8 Key words ............................................................................................78 1.9 Questions for Self- study.....................................................................78

2 Pay off of various Options ........................................................ 79 2.1 Objectives ............................................................................................79 2.2 Introduction.........................................................................................79 2.3 Payoff of long and short call ...............................................................80 2.4 Payoff of long and short put................................................................83 2.5 Risk and premium...............................................................................86 2.6 Illustrations.........................................................................................87 2.7 Summary .............................................................................................89 2.8 Key words ............................................................................................90 2.9 Questions for Self- study.....................................................................90

3 Special applications of Options ................................................ 91 3.1 Objectives ............................................................................................91 3.2 Introduction.........................................................................................91 3.3 Covered Call writing ...........................................................................91 3.4 Protective Put strategy .......................................................................97 3.5 Mimicking and synthetic portfolios....................................................99 3.6 Summary ...........................................................................................102 3.7 Key words ..........................................................................................103


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3.8 Questions for Self- study...................................................................103 4 Options bounds- Calls............................................................. 104

4.1 Objectives ..........................................................................................104 4.2 Introduction.......................................................................................104 4.3 Upper bounds of call prices...............................................................105 4.4 Lower bounds of call prices...............................................................106 4.5 Upper bounds of call prices-American Options................................108 4.6 Lower bounds of call prices-American Options................................109 4.7 Summary of principles of American Options pricing ......................110 4.8 Summary ...........................................................................................111 4.9 Key words ..........................................................................................111 4.10 Questions for Self- study...................................................................112

5 Options bounds -Puts ............................................................. 113 5.1 Objectives ..........................................................................................113 5.2 Introduction.......................................................................................113 5.3 Upper bounds of put prices...............................................................114 5.4 Lower bounds of put prices ...............................................................115 5.5 Upper bounds of put prices-American Options................................116 5.6 Lower bounds of put prices-American Options................................117 5.7 Summary ...........................................................................................119 5.8 Key words ..........................................................................................119 5.9 Questions for Self- study...................................................................119

Module 3 ........................................................................................ 121 ADVANCED TOPICS ON OPTIONS .......................................... 121 1 Option combinations............................................................... 122

1.1 Objectives ..........................................................................................122 1.2 Introduction.......................................................................................122 1.3 Straddle .............................................................................................122 1.4 Strangle .............................................................................................125 1.5 Bull spreads.......................................................................................128 1.6 Bear spread .......................................................................................131 1.7 Butterfly spread ................................................................................134 1.8 Box spread .........................................................................................136 1.9 Summary ...........................................................................................139 1.10 Key words ..........................................................................................139 1.11 Questions for Self- study...................................................................139

2 Principles of Option Pricing – Put call parity. ...................... 141 2.1 Objectives ..........................................................................................141 2.2 Introduction.......................................................................................141 2.3 Some truisms about Options pricing with small illustrations ........142 2.4 Put call parity....................................................................................144 2.5 Exercise of the American Call early .................................................147 2.6 Exercise of the American put early ..................................................150


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2.7 Summary ...........................................................................................151 2.8 Key words ..........................................................................................151 2.9 Questions for Self- study...................................................................152

3 The Binomial model for pricing of Options ........................... 153 3.1 Objectives ..........................................................................................153 3.2 Introduction.......................................................................................153 3.3 Binomial one-period model ...............................................................154 3.4 Binomial two-period model ...............................................................156 3.5 Extension of the principle to greater number of periods.................160 3.6 Summary ...........................................................................................161 3.7 Key words ..........................................................................................162 3.8 Questions for Self- study...................................................................162

4 The Black-Scholes model........................................................ 163 4.1 Objectives ..........................................................................................163 4.2 Introduction.......................................................................................163 4.3 Some preliminary ideas ....................................................................164 4.4 Assumptions under the model ..........................................................165 4.5 The formula .......................................................................................166 4.6 Illustration ........................................................................................166 4.7 The model inputs...............................................................................168 4.8 The Black-Scholes calculator............................................................168 4.9 Impact of variables on Options pricing ............................................169 4.10 Summary ...........................................................................................171 4.11 Key words ..........................................................................................172 4.12 Questions for Self- study...................................................................172

5 Volatility and Implied Volatility from the Black-Scholes model 172

5.1 Objectives ..........................................................................................172 5.2 Introduction.......................................................................................172 5.3 Importance of Volatility and the concept of Implied volatility .......173 5.4 A discussion.......................................................................................174 5.5 Summary ...........................................................................................178 5.6 Key words ..........................................................................................179 5.7 Questions for Self- study...................................................................179

Module 4 ........................................................................................ 180 OTHER DERIVATIVES AND RISK MANAGMENT ................. 180 1 Introduction to Options Greeks and Basic Delta Hedging ... 181

1.1 Objectives ..........................................................................................181 1.2 Introduction.......................................................................................181 1.3 Delta and uses...................................................................................181 1.4 Delta hedging ....................................................................................183 1.5 Gamma, Theta, Vega and Rho..........................................................187 1.6 Summary ...........................................................................................189


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1.7 Key words ..........................................................................................190 1.8 Questions for Self- study...................................................................190

2 Interest Rate Derivatives and Eurodollar Derivatives......... 191 2.1 Objectives ..........................................................................................191 2.2 Introduction.......................................................................................191 2.3 T Bill and T Bond Futures................................................................192 2.4 Hedging with T Bills and T-notes ....................................................193 2.5 Eurodollar Derivatives......................................................................194 2.6 Forward Rate Agreements................................................................195 2.7 Caps ...................................................................................................198 2.8 Floors .................................................................................................199 2.9 Collars................................................................................................200 2.10 Summary ...........................................................................................201 2.11 Key words ..........................................................................................202 2.12 Questions for Self- study...................................................................202

3 Swaps ...................................................................................... 203 3.1 Objectives ..........................................................................................203 3.2 Introduction.......................................................................................203 3.3 Plain Vanilla Interest Rate Swaps...................................................204 3.4 Exploiting disequilibrium in interest quotes – the Spread differential....................................................................................................206 3.5 Currency Swaps ................................................................................209 3.6 Valuing swaps and unwinding .........................................................211 3.7 Collars mimicking swaps ..................................................................213 3.8 Summary ...........................................................................................214 3.9 Key words ..........................................................................................215 3.10 Questions for Self- study...................................................................215

4 Credit Derivatives................................................................... 216 4.1 Objectives ..........................................................................................216 4.2 Introduction.......................................................................................216 4.3 Common Credit Derivatives .............................................................217 4.4 Credit default swap...........................................................................217 4.5 Total Return Swap ............................................................................219 4.6 Collateralized Debt Obligations ( CDOs) .........................................219 4.7 An example of CDO...........................................................................220 4.8 The Indian scenario ..........................................................................221 4.9 Other aspects.....................................................................................223 4.10 Summary ...........................................................................................223 4.11 Key words ..........................................................................................224 4.12 Questions for Self- study...................................................................224

5 Risk Management with Derivatives ...................................... 225 5.1 Objectives ..........................................................................................225 5.2 Introduction.......................................................................................225 5.3 Hedging using Greeks.......................................................................226


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5.4 Delta-Gamma hedging......................................................................231 5.5 Discussion on Hedging Policy ...........................................................233 5.6 Summary ...........................................................................................236 5.7 Key words ..........................................................................................237 5.8 Questions for Self- study...................................................................237

References ..................................................................................... 238


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Module 1

FUTURES AND FORWARDS


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1 Introduction to Derivatives Markets

1.1 Objectives The objectives of this unit are to

• Introduce Derivative instruments

• Briefly look at the common uses and applications of Derivatives

• Briefly look at the trading of Derivatives in the Indian market

1.2 Introduction As financial instruments, Derivatives have become very popular over the last

two decades. While the practice of using Derivatives instruments has been

there for even centuries, the formal application of these instruments in

everyday financial management came about only recently. With the

development of appropriate markets for these securities, a lot of academic

research has also been carried out in their various facets. Understanding

their applications, uses and misuses constitutes an important part of the

study of Financial Management

1.3 Derivatives – meaning and definition Derivatives are instruments in respect of which the trading is carried out as

a right on an underlying asset. In normal trading, an asset is acquired or

sold. When we deal with Derivatives, the asset itself is not traded, but a

right to buy or sell the asset, is traded. Thus a derivative instrument does

not directly result in a trade but gives a right to a person which may

ultimately result in trade. A buyer of a derivative gets a right over the asset

which after or during a particular period of time might result in her buying or

selling the asset.


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A derivative instrument is based first on an underlying asset. The asset may

be a commodity, a stock or a foreign currency. A right is bought either to buy

or sell the underlying the asset after or during a specified time. The price at

which the transaction is to be carried out is also spelt out in the beginning

itself.

1.4 Types of Derivatives There are many types of Derivatives in the market and everyday parlance.

Any transaction that results in a right without actually transacting the asset

becomes a derivative instrument. A brief picture of the common Derivatives

is given below:

Futures and Forwards

Contracts under this category relate to transactions entered into on a given

date to become effective after a specified time frame and subject to payment

at rates determined currently but becoming due after that specified time.

Forwards and Futures are entered into by those who wish to be assured of a

price after a specified time in line with the current price. With prices

fluctuating all the time, it is impossible to predict the price levels after a few

months. A Forward or a Futures contract will ensure that the prices are

frozen upon at the time of entering into the contract and the time frame for

the contract is also firmed up. There are several other aspects to Forwards

and Futures which will be discussed in detail in a later section

Options

Option contracts are a step ahead of the Forwards/Futures contract in that

they result in a right being created without a corresponding obligation. The

buyer of an option contract gets the right without the obligation to either buy

or sell the underlying asset. There is a time frame and a price fixed for the

contract. For the privilege of going ahead with the contract as per her desire,

the option buyer has to pay the seller a premium up front. If, ultimately


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prices do not allow the Options to be exercised, then the premium is the only

loss incurred by the buyer of the Option contract. All the detailed aspects of

an Options contract are covered in a later Module.

Swaps

In a swap transaction the two parties thereto exchange their obligations on

predetermined terms. In its simplest version, two companies having different

obligations of interest payments (with one Company obliged to pay a fixed

rate of interest to its bankers and the other Company having to pay a floating

rate of interest), enter into a contract whereby they exchange their

obligations. This exchange of their obligations results in one Company

getting the fixed interest from the other Company to be used for satisfying its

obligation. In exchange this Company passes on a floating rate of interest to

the counterparty Company to satisfy the latter’s floating interest obligation.

The principal amount to be reckoned for the purpose of calculating the two

interests (called the Notional principal), and the benchmark interest rate to

be used for the purpose of determining the floating rate are decided at the

time of entering into the contract. Swaps are dealt with in detail in a later

module Commodity Derivatives

The most common intuitive use of Derivatives will be in the commodity

segment, where operators fear price rise/fall based on natural weather

conditions. To safeguard their interests these operators can enter into a

buy/sell contract for the required amount of commodity Derivatives.

Typically, like all Derivatives, this does not directly result in the underlying

commodity being traded. Instead, a right or an obligation is established with

respect to the underlying commodity. This type of derivative is also used by

manufacturers and exporters who want to ensure a specified amount of

commodities to meet their business obligations. The principles involved in

these Derivatives are the same as those governing general Options.

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Interest rate Derivatives

Here the parties to a transaction fear a rise or fall in interest rates in the

future and enter into a derivative transaction by which one counterparty

compensates the other when interest rises beyond the agreed rate.

Sometimes these transactions are entered into for getting compensation for

interest rate declines. The notional principal, the benchmark interest rate

and the time of reckoning all are decided at the time of entering into the

contract. Interest Rate Derivatives are covered in a later Module. Credit Derivatives

Bankers and lenders use Credit Derivatives to safeguard themselves against

credit defaults. There are many varieties of these Derivatives involving

sometimes the creation of a third body called a Special Purpose Vehicle.

Credit Derivatives constitute an area of great development in recent years

and many new sophisticated instruments are getting developed by the day.

An introduction to some common Credit Derivatives is given in a later

module.

1.5 Uses of Derivatives Derivatives are used by companies and individuals wanting to cover their

risks. This is facilitated by a counter party who has the motivation to make

profits out of the premium, or is holding a mirror-image opposite position.

Used this way, Derivatives offer an important tool of risk management,

without which companies and individuals would have been exposed to the

vagaries of price fluctuations. However, the use of Derivatives requires skill

in respect of timing, a strategy regarding the extent of coverage and the need

to be consistent in one’s approach. One of the greatest objections to

Derivatives has been that they encourage speculation. In other words, deals

on Derivative contracts can be entered into even by those who do not have a

risky asset position. It can be entered into by speculators betting on a given

price movement or absence of fluctuations. While this in itself may not seem


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to be objectionable, if this practice is carried to disproportionate limits, they

are exposed to huge losses and sometimes bankruptcy. Many companies

have been ruined by over-zealous officials recklessly entering into positions

on Derivatives and taking on enormous risk in the hope of gains on favorable

price movements. Since Derivatives instruments are complex and involve

sophistication in pricing and strategy, it is beyond the non-specialist manager

to comprehend the exact risk that the Company is exposed to because of a

series of derivative transactions. In the process the Company concerned is

exposed to great risk.

While recognizing the possible misuse of these instruments, they are

nevertheless proved to be invaluable in safeguarding Company’s income and

profits. Many companies set up their own strategies regarding the extent of

risk they needed to be covered and correspondingly they enter into

appropriate Derivative transactions for the purpose. This process results in

prevention of unnecessary risk and optimization of profits.

To give an example, an exporter to the United States expects to get $50000 in

3 months from now. She will be happy to have this converted at around

Rs.45 per $. However there is great uncertainty in the foreign exchange

market as to the nature of the possible movement after 3 months.

Fortunately for the exporter a bank is willing to enter into a Forward

contract with her for paying Rs.45 per $ after 3 months on her surrendering

$50000 then. If the exporter enters into this Forward contract, she makes

sure that she will be able to get Rs.2,250,000 ($50000 multiplied by Rs.45)

after 3 months. However, if the rates change to her favor say to Rs.48 per $

she will not be able to take advantage of the favorable movement since she is

already committed to the Forward agreement. A detailed discussion on this

aspect and other related topics are covered in a later unit. There are other


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derivative instruments like Options which enable the trader to have the best

of both worlds, at a price.

1.6 Derivatives in India In India Derivatives have been actively traded over the last decade. The use

of Derivatives in the commodity segment has been existent over several

years, but these were mostly confined to Futures and Forwards transactions.

Options contracts in the stock markets have become very popular in recent

years and have given a new facet to share portfolio management. In the

foreign exchange market, over-the-counter Forwards have been prevalent for

long, but formalized Futures and Options are yet to take shape. Trading of

Interest Rate Derivatives has been formally introduced in the stock

exchanges but these are yet to capture the imagination of the common

investor. Swap transactions have been reported more on a customized one-to-

one basis rather than being taken as formal standardized instruments.

Credit Derivatives have made an entry but are yet to become very popular.

Stock markets find Derivative instruments very useful and portfolio

managers find a number of uses from these for protecting and enhancing

their stock holdings. The rising volumes of Index-based and individual

securities are an indication of their growing popularity. The fact remains,

however, that most of the deals are speculative in nature and are not

necessarily for risk management. But this by itself need not be taken as an

adverse factor, since in most world markets initial uses of derivative

instruments have been basically speculative. Besides, the existence of a large

number of speculators enables the genuine risk manager to put through his

deals comfortably and volumes will not suffer.

The regulation of the Derivatives segments has been handled by the

Securities & Exchange Board of India and the stock exchanges. Strict


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margins and deposits are taken from the trading members to avoid defaults

and payment problems.

1.7 Summary The study of Derivatives involves an approach different from the customary.

In conventional analysis, trading involves buying and selling an asset. In the

Derivatives segment, trading involves not the selling and buying of the asset

itself, but a right on the asset. This right does not carry with it any

obligation and comes at a price called the premium. There are many types of

derivative instruments, the most notable among them being Forwards and

Futures, Options and Swaps. In addition, Interest Rate Derivatives and

Credit Derivatives have become very popular in the US and other countries

in recent years. Derivatives are useful for managing the risk of an

organization. Usually companies develop a strategy for active risk

management using Derivatives. The stock-based Derivatives have become

very popular in India and result in great trading volumes. In India, Forwards

and Futures are in great use in the commodity segment. It is also common to

have Forward contracts in foreign exchange transactions.

1.8 Key words

• Derivatives

• Forwards

• Futures

• Swaps

• Interest Rate Derivatives

• Credit Derivatives

1.9 Questions for Self- study 1) How are Derivative instruments different from regular instruments of

trade?


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2) What are the common type of Derivatives?

3) What categories of investors/traders use Derivatives?

4) Are Derivatives well regulated in India?


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2 Forwards and Futures – a quick look

2.1 Objectives The objectives of this unit are

• To give a general framework of Forwards and Futures contracts

• To understand the benefits of these contracts

• To understand the principle of cost of carry and its uses in practice

2.2 Introduction to Forwards and Futures Forwards and Futures constitute the most basic of derivative instruments.

They are widely used and are quite intuitive in nature. The pricing and

payoff follow a pattern that can be easily understood.

A Forward or Futures contract enables one to enter into an agreement to buy

or sell a specified quantity of the underlying asset after a specified time at a

specified price. In other words, a Forward or Futures contract locks up the

rate of the underlying asset and regardless of the actual rate at the time of

expiry, the deal has to be executed at the rate agreed upon. This

arrangement enables the parties to the contract to lock up their receipts or

payments at convenient levels. However, the disadvantage is that if rates

move in the opposite direction to what is feared, it might turn out to be a

mistake to have entered into the contract. For instance a commodity trader

wishes to sell 500 kgs. of a commodity at Rs.50 per kg. He expects the price

to be steady at this level even after 3 months when the crop will be ready, but

fears that some adverse movements in other sectors might result in a fall in

the price. To safeguard himself he enters into a Forward contract for the

quantity at around Rs.50 per kg. The contract in effect means that he is

obliged to surrender 500 kgs of the commodity after 3 months in exchange of

getting Rs.25000 (500 multiplied by Rs.50). Now, if as feared, the prices fall


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to a level less than Rs.50, the farmer will still get Rs.25000 calculated at

Rs.50 per kg, because that is the agreed rate. However, if the price rises

above Rs.50 to say Rs.60 per kg. the farmer loses on the opportunity profits,

since he is obliged to fulfill his Forward contract at Rs.50 per kg. and will not

be able to participate in the higher profits. Thus, in a Forward/ Futures

contracts, one of the parties to the contract is likely to lose out on the deal in

the final analysis.

To continue an example from the previous unit, an exporter to US expects to

get $.50000 in 3 months time. If the invoicing were made in the Indian

currency, the exporter would not have had any difficulty in estimating her

potential receipt after 3 months. Since the invoicing is in US$, her actual

receipt in terms of Rupees would depend upon the exchange rate at that point

of time. A Forward contract for selling US$ after 3 months at a mutually

acceptable rate would ensure that the exporter gets this rate regardless of the

ultimate actual exchange rate. A Forward contract is thus an agreement to

buy or sell an underlying asset (in this case the US$) for a predestined

quantity, at a predetermined price after a predetermined period. In this case

the buyer of the US$ forward agrees to pay the Indian rupees at the pre-

determined rate. It goes without saying that one of the parties to the

contract will stand to gain more in the final analysis, but what it ensures at

the time of entering into the contract is that the risk element is eliminated.

To take the opposite situation, an importer of goods from the US has to pay

$75000 after 3 months. The invoicing is in $ and so the importer is exposed

to exchange rate risk. The importer is apprehensive that the amount to be

paid may become more in terms of the Indian rupees because of adverse

movements in the foreign exchange market. To ensure that the amount is

frozen, the importer can enter into a Forward agreement to buy $ at a pre-

determined price. At the expiry of the period, the importer pays the agreed


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amount in Rupees for getting $75000. The amount to be paid in Indian

Rupees does not vary with the then prevailing exchange rate. Even if the

exchange rate movement is adverse, the importer is not affected since the

amount to be paid in exchange has been firmed up in advance. However, like

the contract for selling foreign currency seen earlier, here again one of the

parties would lose opportunity gains in the final analysis depending upon the

exchange rates at the time of expiry, but it ensures that the risk is eliminated

at the time of entering into the contract.

Futures contracts work in exactly the same way as the Forwards, except that

they are better regulated. The quantity of the underlying asset that is to be

contracted is in specified lots and the time of expiry is also pre-fixed. For

instance if the importer wants to sell Rs.50000 worth Forward for a period of

3 months, she has to sell this in an exchange contracts corresponding as

nearly as possible to the amount and the horizon needed. Thus if a standard

Futures contract is for say Rs.10000, 5 such contracts have to be sold and if

the contracts expire in 2 months or 6 months, the former is chosen being the

nearest to the horizon needed. There are other structural differences in

Futures as well like the margin requirement, mark-to-market rules and

settlement. These are dealt with in detail later in the Module.

2.3 Basic hedging practices The hedging practice can be formalized through a couple of examples:

A commodity farmer expects 10000 kgs of a commodity to be ready after

harvest in 3 months. The price of the commodity as of now is Rs.2.80 per kg.

The farmer would be happy if the price he obtains is around this level.

However, market economics suggest that the price may take a dip and he

may end up getting only say around Rs.2 per kg.


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A trader in the same area is prepared to get into an agreement with the

farmer to buy the harvest from him at a rate of Rs.2.90 per kg., provided the

farmer commits to the quantity and price today. In other words, the farmer

would be obliged to sell 10000 kgs of the commodity after 3 months at a price

of Rs.2.90 per kg. and the trader would be obliged to buy the quantity at the

price. This will be regardless of what the final price of the commodity is to be

at the end of 3 months.

If he accepts this offer today, the farmer is able to make sure that he gets

Rs.2.90 per kg and he can stop worrying about any possible fall in prices in

the interim. However, he has to continue to worry about obtaining the

harvest of 10000 kgs In case the harvest is not as successful as anticipated

and he ends up having only less than 10000 kgs. ready, he will be forced to

buy from the market the difference in quantity and meet his obligation to the

trader.

As far as the trader is concerned he has ensured that he will get a supply of

10000 kgs. of the commodity at a pre-determined price, and he does not now

have to worry either about changes in prices in the interim, nor about the

availability of the quantity.

This is an example of a short hedge as far as the farmer is concerned and a

long hedge as far as the trader is concerned. In a short hedge the individual

is concerned about fall in prices and sells the commodity in advance at a pre-

determined price. In a long hedge the individual is concerned about the rise

in prices and ensures the price by buying the commodity at the pre-

determined price. In either case the quantity is frozen.

The short hedge has enabled the farmer to reduce his anxiety about the

prices. Now regardless of the actual movement of prices in the market the


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farmer will get Rs.2.90 per kg. If the price at the end turns out to be Rs.2.40

say, he can congratulate himself for having entered into the Forward

agreement, enabling him to force the counterparty to buy from him at 2.90

per kg. On the other hand, if the price rises beyond 2.90 to say Rs.3.20 per

kg. the farmer might feel let down in that he would have been better off

without the Forward contract and would then have been able to sell at

Rs.3.20 per kg. The Forward would force him to sell at Rs.2.90, even though

the actual rate at that time is Rs.3.20. This is the price he pays for ensuring

a minimum amount. He will not be able to participate in upward movement

of prices

The long hedge has enabled the trader to reduce his anxiety about prices.

Now regardless of the actual movement of prices in the market the trader

will have to pay only Rs.2.90 per kg. If the price at the end turns out to be

Rs.3.20 say, he can congratulate himself for having entered into the Forward

agreement, enabling him to force the counterparty to sell to him at 2.90 per

kg. On the other hand, if the price falls to say Rs.2.40 per kg. the trader

might feel that he would have been better off without the Forward contract

and would then have been able to buy from the market at Rs.2.40 per kg. The

Forward would force him to buy at 2.90, even though the actual price at that

time is Rs.2.40. This is the price he pays for ensuring a Forward amount. He

will not be able to take the benefit from downward movement of prices.

In the following example the possible payoff from a short hedge can be seen.

The situation involves selling Forward at Rs.101.51 for 3-month duration. If

the price ends up at Rs.95, he will gain Rs.6.51 on the Futures, but will be

able to sell in the market at Rs.95, making totally Rs.101.51.

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If the price is Rs.103, he will lose Rs.1.49 on the Futures, but can sell at

Rs.103 in the market, thereby making a total of Rs.101.51.

Table I.2.1 Short-hedge payoff (Amount in Rs.)

Price after 3 months

Gain in Forwards

Total proceeds

95 6.51 101.51

96 5.51 101.51

97 4.51 101.51

98 3.51 101.51

99 2.51 101.51

100 1.51 101.51

101 0.51 101.51

102 -0.49 101.51

103 -1.49 101.51

104 -2.49 101.51

105 -3.49 101.51

106 -4.49 101.51

Taking the same situation the position in respect of a long hedge is shown

below. Here again the long hedge has been made at Rs.101.51 for 3-month

duration:

If the price ends up at Rs.97, the long hedge would have to suffer a loss of

Rs.4.51 on the Futures contract, but can get the product at Rs.97 from the

market; thereby the total cost will be Rs.101.51. If the price ends up at

Rs.103, he gains Rs.1.49 on the Futures contract, but has to buy from the

market at Rs.103, resulting in a net position of Rs.101.51.

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Table I.2.2 Long hedge pay off (Amount in Rs.)

Price after 3 months

Gain in Forwards

Total proceeds

97 -4.51 101.51

98 -3.51 101.51

99 -2.51 101.51

100 -1.51 101.51

101 -0.51 101.51

102 0.49 101.51

103 1.49 101.51

104 2.49 101.51

105 3.49 101.51

106 4.49 101.51

2.4 Cost of carry The principles governing fixation of Forward prices are based on interest

rates computation. In Derivatives pricing the universal method is to use

continuous compounding. In other words, the final value based on interest

for a period of 6 months on an investment of Rs.100 at 10% interest is

calculated as 100 multiplied by ℮ raised to the power of the interest rate

multiplied by the time. The ℮ here is the natural logarithm which has a

value of 2.71828. In the above example, the value will be 100 multiplied by ℮

raised to (.10 multiplied by .5). We get 105.13. It may be noted that the

calculations using the ℮ operator can be easily accomplished by a scientific

calculator, or by using the excel function “EXP”. The Excel formula in the

above case will be (=100* EXP (.10*0.5)).

Forward pricing is based on interest rate computation and the principles of

arbitrage. Financial theory has the support of the principle of arbitrage for

various postulates. If the prices as per the postulate do not hold good, it

would be possible for alert operators to buy one type of instrument and sell

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another type of instrument at a risk-free profit. Arbitrage ensures that prices

reach their equilibrium levels in an ideal market.

The theoretical correct price for a Forward contract which has 3 months to go

on a spot price of Rs.40 and an interest rate of 5% can be calculated using the

formula given below:

Forward Price = Spot Price * ℮rt, where

* signifies the multiplication symbol

℮ is the natural logarithm

r is the risk free rate of interest

t is the period of the contract reckoned as a fraction of 1 year

So in the example the Forward price will be:

40*℮. 05*.25 = 40.50

The ideal price for the Forward has to be Rs.40.50.

If the price is greater than 40.50 say Rs.40.75, then an alert operator will sell

the Forward at 40.75 and buy the spot asset at 40. The spot asset will be

bought using borrowed funds which will necessitate an interest payment of

Rs.0.50 for the 3 month period. (Interest is calculated on a principal of Rs.40

for a period of 3 months at an interest rate of 5%, using the continuous

compounding method). On the expiry of the period, the operator will sell the

asset (which he had bought originally in the spot market using borrowed

funds) at Rs.40.75 based on the Forward contract. He will repay Rs.40.50 for

the loan (Rs.40 principal and Rs.0.50 interest) and pocket the difference of

Rs.0.25 as risk-free profit. As more and more operators do this the price will

come back to its equilibrium level of Rs.40.50 for the Forward contract.

If the price is less than Rs.40.50 say Rs.40.25, then an alert operator will sell

the asset in the spot market at Rs.40 and buy the Forward at Rs.40. 25. The

amount got out of selling the spot asset (Rs.40) will be invested in risk-free

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securities earning an interest of 5% for 3 months. i.e. Rs.0.50 (The interest is

calculated on a principal of Rs.40 for a period of 3 months at an interest rate

of 5%, using the continuous compounding method). On the expiry of the

period, the operator will buy the asset based on his Forward contract by

paying Rs.40.25. He will receive Rs.40.50 from his investment (Rs.40 he got

out of sale of the spot asset plus the interest of Rs.0.50 for the 3-month

period), thereby resulting in a net gain of Rs.0.25. As more and more

operators seize on this risk-free opportunity, the prices will reach back the

equilibrium level of Rs.40.50 for the Forward contract.

More aspects of the cost of carry principle and the risk-free arbitrage are

covered in a later unit.

2.5 Differences between Forwards and Futures The following are the broad differences between a Forward and a Futures

contract:

1. A Futures contract is standardized in terms of the quantity per

contract and the time of expiry. A Forward contract, on the other

hand, is customized based on the needs of the two parties to the

contract.

2. There will be no default risk in a Futures contract since it is exchange-

oriented, whereas the possibility of default exists in Forward contract.

In a Futures contract the buyer and the seller do not directly interact

and the exchange is the effective counterparty for each of the dealers

3. A Futures contract will entail a margin for avoidance of default and

this amount has to be remitted from time to time to the exchange

based on extant regulations. In a Forward contract, there is no

standardized margin but this can also be incorporated as a condition to

the contract by the parties concerned.


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4. A Futures contract is monitored on a regular basis by the regulating

authority and hence entails a mark-to-market margin. Thus, if a

trader has bought a Forward contract of 3 months for a commodity at

Rs.50 and if the price is Rs.40 after a week of entering into the

contract, the exchange may require him to pay up the difference of

Rs.10 on each contract. This is because the adverse price movement

might result in ultimate default and the mark-to-market enables the

contract to be scaled up or down to the current market levels. Mark-to-

market margins are generally not insisted upon in a Forward contract.

5. A Futures contract is cash settled. This means that that the final price

of the underlying is compared to the rate agreed upon and the

difference either paid or received from the parties concerned. Actual

delivery of the underlying is not done. A Forward contract, on the

other hand, can be of cash settled or based on physical delivery.

2.6 Summary Forwards and Futures constitute the simplest of derivative instruments.

They are in wide use for risk management. A Forward contract facilitates

the buying or selling of an underlying asset at a pre-determined price after a

specified period. A Futures is similar in operation to Forwards except for

structural variations on account of contractual specifications, margins and

mark-to market. A long Forward contract obliges the buying of the

underlying asset and a short Forward contract obliges the selling of the

underlying asset.

Forwards and Futures are used widely in the hedging of price risks. While

the practice of hedging enables the avoidance of price risk, traders find that

occasionally they lose out on opportunity gains because of price movements

which turn out to be more favorable than expected.

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The pricing of Forwards and Futures follow the cost of carry principle.

According to this, the price at the time of its inception has a definite

relationship with the spot price and is generally represented by the interest

for the period involved. If the price does not conform to this pattern it is

possible to enter to arbitrage and make risk-free profits. That fact that a

number of operators will embark upon this arbitrage will result in the prices

once again coming to the equilibrium levels.

2.7 Key words

• Forwards

• Futures

• Cost of carry

• Hedge

• Arbitrage

2.8 Questions for Self- study 1) A rice farmer is happy to note that the price per kg. for the type of rice

that his farm produces is around Rs.15 now. However, he will get his

crop only after 2 months. He fears that the prices might fall in the

meantime. How can the farmer use Forwards to reduce his risk?

2) Will Forwards always result in profits? Under what circumstances

will a trader feel that he would have been better off without the

Forward?

3) What is margin? Is this applicable only to Futures contracts?

4) If the Forward rate is lower than the rate dictated by the cost of carry

principle, how would an arbitrage be possible?


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3 Hedging with Futures

3.1 Objectives The objectives of this unit are:

• To introduce a framework for hedging

• To look at the risks associated with hedging

• To look at the overall merits and demerits of dealing with Futures

3.2 Introduction The basic hedging model was introduced in the previous unit. Hedging is an

important requirement for all mangers. Based on price fluctuations and

market behavior, managers will tend to hedge their exposures short or long.

We recapitulate the principal factors in respect of hedges below:

1. Hedges using Forwards and Futures can be long or short. A long

hedge is one that goes long the Forwards or Futures (long means buy).

This is entered into by those fearing price rises. By buying the

Forward, they seek to freeze the price’s upper limit. A short hedge is

used by those fearing price falls. A short hedge signifies selling of the

Forward or Futures. By selling the Forward or Futures they seek to

freeze the prices at a level.

2. Hedging is a double-edged sword. While the hedge does offer

protection, it would also mean that if the prices do not move in the

direction feared, one might lose a chance for bonanza profits. Thus in

a long hedge if the price ends up well below the level expected, the

hedge would force the operator to the Forward/Futures price, resulting

in an opportunity loss. In the same way, if the price is greater than

anticipated then the short hedger suffers an opportunity loss.

However, in both the cases, the operators would have frozen upon a


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level of prices which is acceptable to them. It is only the opportunity of

higher gains that they lose in the process.

3. Hedging is a part of the strategic process of companies. They generally

have a policy as to how much of their exposure to hedge and the price

bands at which these should be carried out. Companies tend to leave a

portion of their exposure open. The farmer expects to produce 10000

kgs in 3 months might decide to hedge only 6000 kgs and leave the

remaining 4000 kgs unprotected.

4. Hedging with Forwards and Futures result in almost identical

coverage. However, a Futures contract might entail the payment of

periodic margins and mark-to-market margins, resulting in some

differences in cash flow analysis.

5. The cost of carry principle introduced in the earlier unit needs to be

modified in respect of Futures contract factor because of margins and

deposit money.

6. There will be little default in a Futures contract whereas a Forward,

being a one to one contract might result in defaults.

3.3 Long hedge and Short hedge Let us take the example of an exporter who has sold a commodity to a foreign

country to be delivered three months hence. In order to make reasonable

profits from the deal the exporter has to acquire this commodity from the

domestic market at Rs.245 per unit. The current price is Rs 245 per unit and

the Futures price currently going in the market is Rs.256 per unit. Readers

would have noticed that the Futures price does not conform to the cost of

carry principle introduced in the previous unit. The possible reasons for this

are discussed in the subsequent Chapters. The exporter will buy Futures at

Rs.256 per unit to the extent required.


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Let us assume that the actual price of the commodity has gone up to Rs.290

per unit after three months. This was what the exporter had feared.

However, since he had the foresight to go in for a Futures contract his

interests are protected. Now he gets delivery of the commodity at Rs.256 per

unit which is the agreed price under the Futures contract. The actual price of

Rs.290 does not affect him.

One important difference between Forwards and Futures in respect of final

settlement may be noticed in this context. If the exporter had entered into a

Forward contract he would have got actual delivery by paying Rs.256 per unit

to the counter party. But if he had entered into a Futures contract for his

hedge he will instead be paid the difference between the prevailing

commodity price of Rs.290 per unit and his Futures contract price of Rs.256

per unit. This means that he collects the difference of Rs.24 from the counter

party and buys the commodity in the spot market at Rs.290 per . He was in

any case prepared to pay Rs.256 per unit which is the difference between the

two.

If, however, the price in the spot market ends up at Rs.240 per unit, the

exporter loses Rs.16 on the Futures contract (Rs.256 per unit which is his

contracted price minus the actual spot price of Rs.240). In such a case the

exporter is not able to take advantage of the fall in the prices and still ends

up paying Rs.256 per unit. As we have seen this is the sacrifice he makes for

seeking a hedge using Forwards or Futures.

To take an example a Company fears that the price of its output will come

down. It expects 5000 units of output by the end of next 3 months. The

current output price is Rs.50 per unit. The Futures for the output asset are

currently going at Rs.54 per unit for a 3 month contract. Again the reader


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would have noticed that the Futures price does not strictly conform to the

cost of carry principle. This will be looked at in detail in the next unit.

Similarly, in a short hedge the operator can use Futures by selling the

Futures today. As we have seen the Futures will conform more or less to the

cost of carry principle. After the specified period of contract he will be able to

get the difference between the price at which he sold the Futures, from the

commodity exchange and the actual price in the spot market. If, however, the

spot price in the end happens to be higher than the price at which he sold the

Futures, he will have to pay the difference to the exchange.

Fearing a decline in prices our Company goes for a short hedge by selling the

3 months Futures at Rs.54 per unit for the required quantity. After 3 months

if the final price is say Rs.40 the Company is protected by the Futures

contract. The cash settlement from the Futures contract will give the

Company Rs.14 per unit (Rs.54 contracted in the Futures minus the end spot

price of Rs.40) .The Company will sell the commodity in the market atRs.40

and along with the Rs.14 got from the Futures exchange will be able to pocket

Rs.54 per unit. Here again the procedural difference between a Forward

contract and a Futures contract may be noticed. In the Forward contract the

Company would have been able to sell to the counter party the quantity

contracted at Rs.54 directly. In the Futures contract because it is cash settled

only the difference between the Futures contracted price and the actual price

in the end is handed over to the Company. The actual buying of the

commodity has to be done by the Company through the regular market.

Continuing the example if it so happens that the final price were Rs.70 per

unit the Company will not be able to take advantage of the increase in prices.

Although it will still be able to sell the output at Rs.70 per unit to the

market, it will suffer a loss of Rs.16 (final price of Rs.70 minus contracted


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price of Rs.54) in the Futures market resulting in a net inflow of only Rs.54

per unit. This is the sacrifice for freezing a price using Forwards or Futures.

3.4 Margin requirements for Futures We have seen that Futures are different from Forwards in many aspects.

One such aspect is that Futures are well regulated by stock exchanges.

Every person entering into a Futures contract is actually doing so with the

stock exchange as the counter party

In order to ensure that the parties entering into a Futures contract meet the

stringent requirements of payment, stock exchanges insist on margins.

Margins are amounts required to be paid by dealers in respect of their

Futures positions. There will be initial margins, some special margins

imposed from time to time and mark to market margins.

Mark to market margins result in the dealer having to pay margins specially

for meeting the adverse movement in underlying positions as a result of

changes in spot prices. Some exchanges insist on maintenance margins

which mean that the trader is required to pay additional margin when the

mark to market position falls below a trigger point.

Margin requirements vary from exchange to exchange and sometimes from

time to time. The following illustration shows a typical position and its

impact on the trader based on certain assumptions regarding maintenance

margin and mark to market margin. In the example the initial margin is

Rs.1000 and the maintenance is Rs.750. If the initial margin as adjusted by

adverse mark to market, falls below the Rs.750 level additional margin has to

be remitted by the trader. The table below shows the position:1

1 Example adapted from Derivatives – Valuation and Risk Management, by Dubofski and Miller, Oxford Press


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Table I.3.1 Example of margin liability –Amount in Rs.

Date Sett.Price Initial cash

Mark to Market Equity Maintenance

margin Final cash

Final equity

6-Nov 286.4 1000 -140 860 0 1000 860

7-Nov 288.8 1000 -240 620 380 1380 1000

10-Nov 289 1380 -20 980 1380 980

11-Nov 288.6 1380 40 1020 1380 1020

12-Nov 290.7 1380 -210 810 1380 810

13-Nov 292.8 1380 -210 600 400 1780 1000

14-Nov 292.8 1780 0 1000 1780 1000

17-Nov 292.7 1780 10 1010 1780 1010

18-Nov 295.8 1780 -310 700 300 2080 1000

19-Nov 296.1 2080 -30 970 2080 970

20-Nov 297.1 2080 -100 870 2080 870

21-Nov 296.4 2080 70 940

On 5th Nov when the contract was entered into the price was Rs.285. The

trader had taken a short position in Futures and has paid an initial margin of

Rs.1000. On 6th Nov the price fell to Rs.286.40 resulting in his effective

margin amount falling to Rs.860, as shown under column “Equity”. The next

day the spot price was Rs.288.80 resulting in a further depletion of Rs.240.

As shown under the Equity column the margin now is only Rs.620. Since the

maintenance margin is Rs.750 and the Equity has fallen below that level the

trader is required to replenish the margin to the original level of Rs.1000.

This entails a payment of Rs.380 from his side as shown under the column

“Maintenance margin”. This procedure goes on till the end of the contract.

The last two columns show the effective Equity position and the margin

position from time to time.

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3.5 Basis risk Basis risk refers to the changing difference between the spot asset prices and

the Futures prices. Basis risk has greater application in Futures contracts

because here the contracts do not expire at the time when the trader

requires. In a Forward contract it is possible to customize the contract to the

appropriate time frame.

Basis refers to the difference between the Spot price of the asset and the

Futures price of the asset. Let us work with the following symbols

S1 = Spot at time t1

S2 = Spot at time t2

F1 = Futures at time t1

F2 = Futures at time t2

B1 = Basis at time t1

B2 = Basis at time t2

Suppose the spot price of an asset at inception was Rs.2.50 and the Futures

price at that time was Rs.2.20. After 3 months, the spot price becomes 2.00

and the Futures price at that time is Rs.1.90. Here the basis at the beginning

(B1) is - 0.30 and the basis at the end (B2) is - 0.10

When the spot increases by more than the Futures, Basis strengthens

When the Futures increases by more than the Spot the Basis weakens

In a hedge the final price =

S2 + F1 – F2

This is equal to F1 + B2

F1 is known at inception

If we can correctly estimate B2, the hedge can be perfect


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The above situation can be elaborated a little more. At the time when the

contract is entered into there is a spot price for the asset and a corresponding

Futures price. Normally, the Futures will conform to the cost of carry

principle and can be determined fairly accurately. But the actual Futures

price may not sometimes conform to the cost of carry rule for various reasons

discussed in the next unit. The difference between the spot and the Futures

price at the start is the basis at the beginning. As time goes on, the spot

price changes and the Futures price also changes. At the expiry time, which

is the next crucial juncture, the Spot and the Futures are at another level of

difference between each other. Occasionally, the difference levels will remain

identical. Sometimes the differences go up or come down. When the

difference goes up (Spot – Futures goes up), the basis is said to have

strengthened. When the difference comes down (Spot-Futures comes down)

the basis is said to have weakened.

A short hedger expects prices to come down and hence has sold the Futures.

At expiry, he settles his position with the Futures price at the end. If the

Futures price at the end is in line with the Spot at the end in exactly the

same level as the difference at the beginning, the basis is the same. In such a

case, the hedge will be perfect. He will realize exactly what he sought out to

do. For instance, the spot at the beginning was Rs.100, and the Futures were

102. He had sold Futures at Rs.102. At the end the spot is Rs.85 and the

Futures is Rs.87. Now he realizes Rs.15 (the difference between 102 and 87).

Suppose the basis had strengthened, the Futures price would have been less

than Rs.87, say Rs.86. the basis at the beginning was (-2) and it has now

become (-1). The (-2) difference has become (-1). The hedger would make

Rs.16 (difference between 102 and 86). Here the short hedger has benefited

from the basis strengthening.

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On the other hand, if the basis had weakened and if the final Futures price is

88, the hedger would have made only Rs.14 (102-88). He would have lost out

of the basis weakening. The basis originally was (-2) and it has become (-3).

The converse is true of a long hedge. Here the hedger had bought the

Futures. If the spot initially was Rs.200 and the corresponding Futures was

Rs.203 at inception. At expiry the spot is Rs.240 and the Futures Rs.243.

Here the basis remains the same at the start and at the end. So there is no

gain or loss from basis risk. The hedger would get Rs.40 (243-203) from the

long hedge. If the basis had strengthened the Futures would be Rs.242 for

instance. The basis originally was -3 and it has become -2. Here the hedger

would get only Rs.39 (242-203). He has lost on the basis strengthening. If,

the basis had weakened and the Futures price at the end was say 244, he

would have gained Rs.41 (244-203). Here the original basis was (-3) and the

final basis was (-4), so the basis had weakened.

Basis risk arises because the hedge position cannot possibly go on up to the

last date desired. If the horizon required by the hedger exactly conforms to

the horizon of the Futures contract then there will be only negligible basis

risks. The principle of convergence which avoids basis risks is discussed in

detail in the next unit.

A change in basis can result from a change in the risk free rate of interest,

change in floating funds, change in the availability position of the assets and

a phenomenon called convenience yield.

Cross hedges have greater basis risks because the underlying assets are

different.


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Even though basis risk is a very real factor, the fact that un-hedged positions

carry greater risks means that former can be ignored as a factor in hedging.

3.6 Cross hedging In all the examples that we have discussed so far there were ready-made

Forward/Futures contracts available for the asset concerned. The Company,

trader or operator was able to directly use the Forwards/Futures contract for

his hedging purposes. In actual practice we may not have ready

Forward/Futures contract available for an asset of our interest.

In such cases the Company has to identify another asset which is covered by

going Forward/Futures contracts. Having identified the asset which is similar

in nature to the asset on which the Company wants a hedge, the extent of the

relationship should be estimated. There is no hard and fast rule for this

estimation.

To take an example a farmer expects 500 kgs of onion in the next three

months. He would be happy to be able to sell the output at around the price of

Rs.15 prevalent today. Unfortunately for him there is no Forward/Futures

contract which would have enabled him to go in for a short hedge. He

however, notices that there are running contracts of Forwards/Futures in

respect of potatoes. Based upon his past experience he feels that the price

movements of potatoes and onions are perfectly co related. In other words a

Re 1 increase in the price of onion is generally concurrent with a Re.1

increase in the price of potato. So he performs his short hedge using his

Futures potato contracts. He will short hedge by selling 500 kgs of potatoes in

the Futures market at Rs.15 per kg. If after 3 months the price of potatoes

falls to say Rs.12 he will be able to enforce his Forward/Futures contract by

being able to get a selling price of Rs.15 per kg. What this effectively means is

he will be able to buy from the spot market at Rs.12 per kg then and sell it at


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Rs.15 to Forward/Futures market at Rs.15 per kg thereby making a gain of

Rs.3 per kg. Corresponding to the decline in potato prices the onion price

would have also be fallen to around Rs.12 per kg. But the farmer is protected

in the sense that he will be able to sell the onions at Rs 12 per kg in the

market and get Rs.3 per kg as gains from the potato future contract.

Effectively he is able to get Rs.15 per kg which was what he wanted in the

first place.

The practice of hedging with a different asset to the asset of interest to the

hedger is called cross hedging. The following aspects may be noted in respect

of cross hedges.

1. The first step in a cross hedge is to identify another asset similar in

nature to the asset of interest.

2. The exact extent of relationship should be estimated based on past

experience or historical records. In the above example onions and

potatoes were estimated to be perfectly correlated.

3. If there exists a relationship but the extent of correlation estimated is

less than 1, the Company must make adjustments to the quantity to be

hedged so that the hedge becomes near perfect.

4. If in the above example a Re1 change in onion prices is generally

estimated to accompany with a Re.1.50 change in potato prices, the

extent of contracts required in the Futures market is calculated by

Beta approach. Here the Beta of onions to potatoes is 0.67(1 divided by

1.50). Therefore if 500kgs of onions are to be covered, approximately

333 contracts of potatoes will be needed.(0.67 multiplied by 500)

5. The principle here is that if onion prices fall by Rs.2 potatoes prices

would have to fall by Rs.3. The 333 kgs of potato Futures contract will

effectively safeguard the fall for 500kgs of onions (333kgs of potatoes


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multiplied by Rs.3 is approximately equal to 500 kgs of onions

multiplied by 2).

6. Basis risk constitutes an important disadvantage of cross hedges

7. Besides the time horizon of the Futures contract may not conform to

the required time horizon of the Company. For instance in the above

example potato Futures contract may not be available for 3 months but

may be available for only 6 months. Since the farmer requires coverage

only for 3 months in respect of his onions output a suitable Futures

contract will be difficult to find.

3.7 Summary Hedging constitutes the most important use of Futures. Hedges are either

long or short. In a long hedge a Futures contract is bought and in a short

hedge a Futures contract is sold. Hedging is a double-edged sword. It offers

protection for adverse movement of prices but prevents the hedger from

participating in extreme favorable movements.

When a Futures contract is not available for a specific underlying asset it

may become necessary to hedge with another asset similar in nature to the

desired underlying. This is called cross hedge. Here the hedger will have to

estimate the extent of change likely to occur in the second asset for a change

in the underlying asset. Accordingly the desired number of contracts is

entered into on the second asset - either as a short hedge or a long edge. If

the estimated movements of the two assets are in conformity with

expectations, a cross hedge will perform as efficiently as a regular hedge.

In entering into transactions with Futures a trader is exposed to basis risk.

A basis risk refers to the difference between the spot and the Futures prices

at the beginning and the changes thereto at the end. A short hedger benefits


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from strengthening of the basis and a long hedger benefits from the

weakening of the basis.

Futures contracts entail margins as imposed by stock exchanges. The

practices of levying margins are different in various exchanges and usually

have components like an initial margin, some maintenance margin and

mark-to-market margins. Traders are required to keep paying the necessary

amounts to the exchange. This procedure enables the exchange to manage

the risk and prevent default. It must be remembered that in Futures

contracts, the counterparty is always the Exchange.

3.8 Key words

• Hedging

• Basis

• Cross-hedge

• Strengthening of basis

• Weakening of basis

• Margin

• Mark-to-market

3.9 Questions for Self- study 1) What is basis risk? Is it important in hedging?

2) In a long hedge, if the price of the underlying falls, what will be the

nature of the final payoff for the hedger?

3) How is the appropriate asset chosen in a cross-hedge, where the

desired underlying is not traded in the market?

4) How are margins levied by stock exchanges? What is the role of mark-

to-market in this?


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4 Pricing of Futures and arbitrage conditions


• To understand the principle of arbitrage and how it ensures a fair

Futures price

• To look at the general rules and exceptions to these rules in respect of

Futures pricing

• To understand the evidence from the market on the principle of cost of

carry

4.2 Introduction The basic principle of cost of carry and its application in Forward/Futures

pricing was introduced in an earlier unit. Operators in a stock exchange

should be well-versed with principles governing the pricing of Derivatives, to

enable them to use these in the right manner. These postulates in pricing

are intricately connected to basis risk and other economic imbalances that

might be existent.

Many economic conditions exist which prevent absolute perfection of

markets. These imperfections result in the absence of perfect arbitrage and

consequent in equilibrium in pricing

Lastly, an efficient market will necessarily want the Futures to be priced as

close to the cost of carry principle as possible, so that other strategies like the

calendar spread and rolling the hedge Forward can be practiced.

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4.3 Basic pricing principles We have seen the following equation for Forwards pricing. The same applies

to Futures pricing as well, with modifications to provide for margin

remittance.

Futures Price = Spot Price * ℮rt, where

*signifies the multiplication symbol


r is the risk free rate of interest : t is time to maturity

The basic model applies where there is no expected income from the

underlying during the tenure of the contract.

In case a specific dividend is expected from the asset during the period of the

contract, the formula has to be modified as follows:

Futures Price = (Spot Price- Expected Dividend) * ℮rt, where



r is the risk free rate of interest : t is time to maturity

The principle behind the modification is that if there is to be a dividend, then

the cost of carry will be lesser by that amount. The amount required for

investment will be lower by the extent of dividend.

Sometimes, instead of a specific dividend the underlying has an expected

dividend yield. This is particularly applicable for Index based Futures, which

is the subject matter of the next unit. The formula for the Futures price will

then be:

Futures Price = Spot Price * ℮(r-y) t, where



r is the risk free rate of interest: t is time to maturity

y is the expected dividend yield.

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Here the dividend is not a specified amount, but a specified yield, and so the

adjustment is made to the rate of interest. The cost of carry principle works

on the principle of interest carry and the interest levy will be lower when

there is a specific income yield

The impact of the time and rate of interest on the cost of carry can be seen

with an example.

Assuming the asset price today is Rs.200, the time to expiry of the Futures

contract is 3 months and the interest rate to be 6% p.a, the Futures price

should be Rs.203.02, based on the cost of carry principle. This is calculated

as

200 * ℮ (0.06 * 0.25)

The time has been taken as 0.25 (3 months), and the rate of interest in

decimals is 0.06.

The impact of changes in interest rates and time to expiry on the Futures

contract is shown in the following table:

The table shows the relative impact of interest rates and the time to expiry

(both expressed in decimals). As the interest rates go up, keeping the time

constant, the Futures price goes up. This is intuitive in that a greater rate of

interest results in greater cost of carry. Similarly, as time goes up, keeping

the interest rate constant, the cost of carry goes up. This again is intuitive in

that the greater the time period, the greater the carry element.


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Table I.4.1 Cost of carry and formula-based prices Interest

Time

0.25 0.35 0.45

6.00% 203.0226 204.2444 205.4736

6.25% 203.1495 204.4232 205.7048

6.50% 203.2765 204.6022 205.9364

6.75% 203.4036 204.7813 206.1682

7.00% 203.5308 204.9605 206.4003

7.25% 203.6581 205.1399 206.6326

7.50% 203.7854 205.3195 206.8652

7.75% 203.9128 205.4992 207.0981

8.00% 204.0403 205.6791 207.3312

8.25% 204.1678 205.8592 207.5645

8.50% 204.2955 206.0394 207.7982

8.75% 204.4232 206.2198 208.0321

9.00% 204.551 206.4003 208.2663

In Futures contracts, the cost of carry has to factor in the possible margins

and the interest thereon into the calculations.

Another factor that has occasionally resulted in differences is the rate of

interest to be reckoned for calculation. Academics are divided as to the most

relevant rate of interest for this purpose. The broad consensus is that the

Treasury Bill rate, which is risk-free, should be reckoned for calculation. The

logic behind this is that the theoretical prices are held in a place by arbitrage

and unless the arbitrage is risk-free, it cannot be universal. A risk-free

arbitrage can have only a cost of carry of the risk-free rate.

4.4 Arbitrage opportunities The possibility of arbitrage makes the theoretical price rigid.

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Let us take the case of an asset having a price of Rs.100. The Futures to

expire 3 months from date will have a price of Rs.101.51, if the interest rate

is 6%. (100 * ℮ (0.06 * 0.25)).

Suppose the price is only Rs.101, operators will seize the opportunity to buy

Futures at 101 and sell the spot asset at Rs.100. In 3 months they will earn

an interest of Rs.0.51 on the sale proceeds of the spot asset, and use Rs.101 of

this to buy back the asset. The difference of Rs.0.51 is their risk-free profit.

There will be a gain regardless of the final price of the asset. This is shown

in the following table:

Table I.4.2 Arbitrage when actual Futures is less than theoretical price If final price is

gain on spot

gain on Futures

total gain interest net

gain -2 1 -1 1.51 0.51

99 1 -2 -1 1.51 0.51

100 0 -1 -1 1.51 0.51

101 -1 0 -1 1.51 0.51

102 -2 1 -1 1.51 0.51

103 -3 2 -1 1.51 0.51

104 -4 3 -1 1.51 0.51

105 -5 4 -1 1.51 0.51

106 -6 5 -1 1.51 0.51

107 -7 6 -1 1.51 0.51

108 -8 7 -1 1.51 0.51

109 -9 8 -1 1.51 0.51

110 -10 9 -1 1.51 0.51

As more and more operators take this opportunity, the price of the Futures

will get automatically adjusted to the theoretical level of Rs.101.51.

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In the same way, if the final price is Rs.102 (greater than the theoretical

level), alert operators will sell Futures and buy the asset in the spot market

at Rs.100. They will borrow Rs.100 for buying the asset spot. They will incur

an interest of Rs.1.51 over the 3-month period for the borrowed money. They

will, however, realize Rs.102 from the sale of The Futures. This will be

available to them after 3 months, and they will use Rs.101.51 of that for

repaying the interest and principal of the loan, thereby pocketing the

difference as risk-free profits. The risk-free profits will be available to them

regardless of the final spot price, as shown in the following table:

Table I.4.3 Arbitrage when actual Futures is greater than theoretical price End spot

gain spot gain F tot gain int. loss net

gain 99 -1 3 2 1.511306 0.49

100 0 2 2 1.511306 0.49

101 1 1 2 1.511306 0.49

102 2 0 2 1.511306 0.49

103 3 -1 2 1.511306 0.49

104 4 -2 2 1.511306 0.49

105 5 -3 2 1.511306 0.49

106 6 -4 2 1.511306 0.49

107 7 -5 2 1.511306 0.49

108 8 -6 2 1.511306 0.49

109 9 -7 2 1.511306 0.49

110 10 -8 2 1.511306 0.49

As more and more operators take this opportunity, the price of the Futures

will get automatically adjusted to the theoretical level of Rs.101.51.

While the possibility of arbitrage ought to restore the Futures prices to their

correct theoretical position, the following factors need to be considered:


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1. For arbitrage to be possible, the information regarding Futures prices

should be constantly available to all possible dealers. In India, stock

markets work on an online real-time basis andhence the information

will be readily available. In the commodity segment and the foreign

exchange segment, information may not be that easily available and

this factor will create the difficulty for arbitrage.

2. When the actual Futures prices are greater than the theoretical level,

the dealer will seek to sell Futures and buy the Spot. For buying the

Spot, he may need borrowed money. Depending upon the economic

situation in the country, money supply may or may not be freely

available. If it is available only at high rates of interest, the arbitrage

may not work out to any benefit.

3. The basic assumption under the principle of cost of carry is that both

borrowing and lending can be done at the risk-free rate of interest.

This assumption itself is not far-fetched since any model requires a

stable assumption to proceed. The Capital Assets Pricing Model and

the Modigliani-Miller propositions all assume this. However, the

availability of ready funds at this rate is crucial.

4. When the actual Futures price is less than theoretical level, the dealer

will seek to buy Futures and sell the Spot. Many times this may not be

possible because of the absence of ready stock of the Spot to sell.

Shortage of delivery position in the market might result in this

difference not being exploited.

5. Sometimes, the phenomenon of convenience yield prevents selling the

Spot even though an arbitrage opportunity exists. The holder of the

Spot might feel inclined to hold on to the stock for the convenience of

having the stock ready, rather than selling and getting back the stock

after some time. This is particularly true of export traders who like to

keep their stock ready for possible exports in the future.

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6. Sometimes regulatory provisions may create obstacles to smooth

arbitrage. Prohibition of short selling and disclosure of open positions

might result in arbitrage opportunities not being exploited. This could

also be because of the need to have a minimum quantity of buy or sell

in the Futures segment. The amount required as margins may not be

easily forthcoming and this again could prevent arbitrage.

4.5 Empirical evidence on cost of carry Empirical evidence on the cost of carry principle has not been consistent.

Most studies have shown that cost of carry does not perfectly hold in markets

over a length of time. Many economic reasons including those listed above

are given for the phenomenon.

A typical study on the behavior of Futures prices to changing spot prices

would look at spot rates at various dates and compare these with the

corresponding Futures rates. Over a period of time, the interest rate taken

by the market can be intuitively understood. When this rate is steady over a

period of time, we can gather that the cost of carry principle holds and that

the interest rate attributed is rigid. However, studies of this nature have

found the cost of carry varying from week to week and sometimes even day to

day.

Studies have also found the existence of backwardation in Futures prices.

Backwardation refers to the Futures price being lower than the Spot price.

This is counter-intuitive to the cost of carry principle. But the predominant

weight of the other economic factors and market imperfections do lead to this

from time to time.


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4.6 Rolling the hedge forward In all the examples discussed so far the hedging was successful because the

time horizon of the hedger matched substantially with the time horizon of the

Futures contract. There will be a great deal of difficulty if this matching is

not perfect.

Let us take the case of a trader wanting a long hedge for a period of 6

months. He fears that the prices will go up for the commodity in 6 months

and wants to lock in the price at a level. He understands that a Futures

contract on the commodity will be the ideal answer to his problem. However,

he finds that Futures contracts exist only for a 3-month horizon, while his

requirement is for 6 months.

In such a case, he can go in for rolling the hedge forward. We can look at this

phenomenon through an example (all figures in Rs.):

The rolling of the hedge forward involves first taking Futures of a suitable

duration and just before its expiry, squaring it off and going in for a new

Futures contract. If the horizon is not met even after the second set of

Futures, the process is repeated as many number of times as makes the

horizon match.

In the above example it would not have mattered if the Spot price at the end

of 3 months had actually been less than the original. The square off will be

done at the spot price at end of 3 months ( the Futures will be very near this

spot price near expiry because of the principle of convergence and low cost of

carry) and the new Futures will be taken up at around the same price.


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Table I.4.4 Rolling the hedge forward (Amount in Rs.) Spot price 100

Risk-free rate of

interest 6%

Time 3 months 0.25

Futures price 101.51

The trader goes in for a Long hedge buying Futures at 101.51,

to safeguard against rise in prices

The requirement being for 6 months the hedge is not

complete even after 3 months.

After 3 months

Let us say the

Spot is 110

At this time, the Futures for other 3-months will be available

at Rs.111.66. The trader will square off his original position

at the converged spot price.

Long new Futures 111.66

The trader’s gain (110-

101.51) on the old Futures 8.49

After 6 months

Spot is say 120

Gain (120-111.66) on new

Futures 8.34

Total gain 16.83

Spot trader was ready to pay 100

Interest saved 3.05

Total he can now pay 119.87

actual spot 120

Sometimes tracking errors can occur and there could be resultant differences

in the prices of square off and new Futures. So the gains listed above may not

always occur accurately. But hedging is all about approximation and the

trader will continue to be covered by and large.


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One great disadvantage of rolling the hedge forward is the high amount of

transaction costs that are likely to be incurred in the process. Ultimately, like

a single-period hedge, a rolling of hedge is also a double-edged sword. If the

price movements are not as anticipated, the potential for participation in

these will be foregone. However, the rolling process can be reviewed at the

end of each interval.

4.7 Summary The principles of hedging for Forwards and Futures are substantially the

same. In Futures, we also have to reckon the question of margins. A long

hedge involves going in for buying the Futures contract fearing a price rise.

A short hedge involves going in for selling the Futures contract fearing a

price fall. In either case, the hedge is based on the going Futures prices,

which, in turn is expected to conform to the cost of carry principle.

The pricing of Futures is basically by the Cost of carry principle. Continuous

compounding is used in calculating interest rates and for this the natural

logarithm is taken as the basis. In case the asset has a known return during

the period of the contract, the amount of such known income is deducted from

the Spot price today to calculate the cost of carry. The principle behind this

is that the known income reduces the amount involved in the spot

investment. In the same way, sometimes, it is a known dividend yield that

comes about and not a fixed known income. Here, the dividend yield is

deducted from the risk free rate in calculating the cost of carry.

The principle of Futures pricing is based on arbitrage. If equilibrium does

not hold then alert investors will invoke arbitrage and reap risk-free profits.

As the number of such operators goes up, the prices come back to the

equilibrium level. However, there are certain economic factors which may


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prevent perfect arbitrage. These could also be sometimes regulatory in

nature. In such cases the Futures prices will not conform to the cost of carry

principle and what is more, may sometimes go into backwardation.

Empirical evidence on these prices has not established that cost of carry as a

principle will always hold good. In fact, the evidence has been weighed in

favor of its not holding most of the time.

When the horizon of the hedger does not match the horizon of a Futures

contract, the hedge can be rolled forward. This involves taking a Futures

shorter than the required horizon at first and then shifting this to other

Futures on the expiry of the first. By and large this will result in a perfect

hedge. However, high transaction costs and occasional tracking errors might

result in some losses. Like a regular hedge the rolling hedge will also be able

to ensure around the agreed price regardless of the movement of spot prices.

By the same token, any unforeseen profits cannot be participated in because

a Futures contract is a commitment at a particular level.

4.8 Key words

• Long and Short hedges

• Cost of carry

• Continuous compounding

• Arbitrage

• Rolling the hedge Forward

• Convenience Yield

• Backwardation

4.9 Questions for Self- study 1) How is the theoretical Futures price computed when it is known that

the asset is expected to yield a dividend of 5% during the period of the

Futures contract?


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2) Does the principle of rolling the hedge Forward always result in a

perfect hedge?

3) What circumstances prevent the Futures prices from following the cost

of carry principle?

4) Explain the intuition behind using the risk-free rate for calculating the

Futures prices based on cost of carry.


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5 Stock Index Futures


• To introduce the concept of Index Futures

• To see the applicability of pricing theories to Index Futures

• To see some of the higher uses of Index Futures

• To understand the principle of cross hedges as applicable to Index

Futures.

5.2 Introduction Several years after Futures trading got into full swing in both the NSE and

the BSE, the investing community in India does not appear to be fully aware

of all the possibilities that these offer. This unit attempts to draw attention

to common uses and certain possibilities of sophisticated uses of Index

Futures. At a given point of time, there are three Futures being traded in

each exchange – one expiring on the last Thursday of the third month from

the date of the trade, another expiring on the last Thursday of the second

month from the date of the trade and yet another one expiring on the last

Thursday of the month succeeding the date of the trade. The 3-month

Futures of today will thus become the 2-month Futures after one-month and

a 1-month Futures after 2 months. The permitted lot size of S&P CNX Nifty

Futures contracts is 200 and multiples thereof. The spot price at the expiry

of the last trading day will be reckoned as the converged Futures price for

settlement purposes. In other words, this means that as on the last date of

trading of the Index Futures, the Futures price will equal the spot price. For

instance, if Futures are expiring on 27th July 2000, the final rate for this will

be exactly equal to the spot Index rate on that day. As a corollary, it means

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that the differences between the Futures price and the spot price will narrow

down as the expiry date approaches.

Details regarding the rules of trading and settlement can be had from the

NSE web site - www.nse-india.com

5.3 Construction of stock indices Stock indices are indicators of the market. There are many types of indices.

Broadly these can be classified into price-weighted and market-weighted.

The market-weighted indices are the more common and generally considered

to be less prone to wild fluctuations. The principle behind the construction of

a market-priced Index involves first identifying a given portfolio of stocks

that have the maximum market identification. The exact number could be

anything and in India the NIFTY Index uses 50 as this number, while the

BSE sensex takes 30 as the corresponding number.

The most popular stock Index in India is the S&P Nifty Index and the Junior

Nifty Index, managed by the National Stock Exchange, and is taken as the

basis. The S & P Nifty Index is based on the following principles:

The stocks in the Index are among the top market capitalized companies in

the country. The stocks in the Index are liquid by the “impact cost” criterion.

The liquidity of stocks at the given price level is tested by relative price

determination. The market impact cost tends to accurately reflect costs faced

when actually trading an Index.

In order to qualify for S&P CNX Nifty, the criteria is that it has to reliably

have a market impact cost of below 1.5%, when doing S &P CNX Nifty trades

of 5 million rupees. Among the shares that get so qualified, the 50 companies

with the maximum market capitalization go into the Index. The next 50

companies with the same criteria go into the JUNIOR NIFTY category.

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NIFTY’s structure is based on a paper by Shah and Thomas (1998)2. The

paper looks at the question of illiquidity of market indices and how this can

influence Index construction. The authors say that the evidence against

hedging effectiveness and the evidence on the impact cost of alternative Index

sets led to the choice of NSE Index in 1996. Two rules govern the ongoing

modifications to the Index set:

• A minimum liquidity filter is applied. This is based on impact costs in

transitions involving trades of Rs.5 million if the particular security is

in the Index. A security is considered eligible if the impact cost is 1.5%

in 90% of the snapshots in the last six months.

• An eligible security is admitted into the Index displacing the smallest

security in the Index if the incoming security is at least twice the

market capitalization of the outgoing security, but at the same time

prevents excessive changes.

5.4 Uses and applications of stock Index Futures The use of stock Futures specifically arises out of the principle of

convergence. The spot prices and the Futures prices have a relationship

based on the cost of carry. The prices converge on expiry. This and other

aspects of Stock Index Futures, as distinguished from regular Futures are

given below:

• There are three Futures going in the market at a given point of time.

Theoretically, these three Futures must be all following the cost of

carry principle and hence the longest Futures (3-month Futures) must

have a price proportionately higher than the 2-month Futures and the

1-month Futures.

2 Shah Ajay and Thomas Susan, Market Microstructure considerations in Index constructions, 1998

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• As maturity draws near, the cost of carry keeps coming down until, on

the date of expiration of the Futures the price converges with the Spot

price. Here there will be no cost of carry and hence the Spot price and

the Futures prices will be equal.

• If the investor is planning an Index portfolio (portfolio of stocks in

proportion to those in the Index), she will be able to hedge her portfolio

by buying or selling the appropriate number of Futures contracts.

Generally, the regulatory authorities fix a minimum amount of

Futures that needs to be bought as one lot. This figure changes from

time to time and from exchange to exchange. The current rules

followed in the National Stock Exchange can be seen from their

website3

• Further, if the holding period matches with the horizon of the Futures,

a perfect hedge is possible

• Additionally, Index Futures offer Cross hedges and rolling the hedge

forward in a more structured manner. Besides, there are several

advanced uses for Index Futures discussed below.

5.5 Hedging with stock Futures

• Futures prices have a definite relationship with the spot prices. If this

relationship does not hold it will be theoretically possible to have a

risk-free arbitrage between the spot and the Futures. This

relationship stipulates that in normal conditions, the Future prices

must be more than the spot price by a “cost of carry”, which are the

interest charges for holding the spot to the duration of the Futures

minus any dividend that holding the spot will entitle the holder. Thus,

if the spot price on a given day is “x”, the Futures price on that day will

be “x” plus the interest for holding the spot to the duration of the

3 www. nse.co.in, giving full details of settlement and margin positions for Futures

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Futures contract, minus any pre-determinable dividend or dividend

yield on the spot.

• Suppose this relationship does not hold, the Futures are incorrectly

priced and this gives rise to a theoretically risk-free arbitrage. For

instance, if the Futures price is lower than the theoretical price

described above, it will be worthwhile for a share dealer to buy the

Futures and simultaneously sell the spot, The proceeds from the sale of

the spot could then be invested in risk-free securities and at the end of

the tenure when the Futures contract expires, he would buy the stock

back on the strength of the Futures contract at a cost less than what

his investment has yielded. He thus makes a clean risk-less profit. As

more and more investors like him do this Futures prices will find its

correct level vis-a vis the spot. This is illustrated in the box alongside

Suppose the spot Index price is Rs.1250 and the one-month Futures are

trading at 1280. Assuming that the risk-free rate of interest will yield an

interest of Rs.10 on Rs.1250, the theoretical Futures price ignoring dividends

is Rs.1260. Because the actual price is Rs.1280, dealers in the market can

sell the Futures at Rs.1280 and buy the spot at Rs.1250. After one month, on

the convergence date, the spot and the Futures will be priced identically by

rule. At that time our dealer would square up the Futures position by buying

back and simultaneously sell in the spot market. The outcome of his deals

will be as follows. It has been assumed that the converged price after 1

month is Rs.1290. It does not make any difference what that price is,

because, both the spot and the Futures will be priced identically then.

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The whole process results in

a risk-free return of Rs.20.

(The difference between the

buy and sell deals on

Futures results in a loss of

Rs.10, and the cost of carry

loss on borrowing is Rs.10,

but the stock gains are

Rs.40). As more and more

dealers seize this

opportunity, the Futures

price will adjust itself to the

theoretically correct price.

For this to succeed the dealer must either have the stock of the Index stocks

ready with him for sale, or he should be in a position to borrow the stocks for

a short period.

Now let us take the converse case. If the actual Futures price were higher

than the theoretical Futures price, the trader would borrow money and buy

the spot and sell the Futures. At the expiry of the Futures contract, he would

offer the stocks in delivery and collect the proceeds, which will be higher than

the price of the stock plus the interest. Again, he makes a risk-free profit. As

more and more people do it, the Futures price will find its correct level, vis-a

vis the spot.

Suppose the spot is Rs.1250 and the Futures are Rs.1220. Assuming a

theoretical cost of carry of Rs.10 on the Rs.1250, the Futures price ought to be

TODAY

Sale proceeds of Futures Rs. 1280 (+)

Price of buying stock Rs. 1250 (-)

AFTER ONE MONTH

Sale of spot Rs. 1290 (+)

Buy back of Futures Rs. 1290 (-)

Cost of carry Rs. 10 (-)


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Rs.1260. Since it is trading at a discount to this, dealers will buy the Futures

and sell in the spot segment. For doing this, they would require ready

delivery of the stocks constituting the Index, in the proportion as near as can

be to the Index. Alternatively, they must have access to stock borrowing

facility. Otherwise, they could be a portfolio holder seeking to take

advantage of the disparity. After one month, convergence of prices takes

place. Then the Index Futures position is squared off by selling and at the

same rate the stock is bought back.

This results in a risk-free gain of

Rs.40. (The gains of Futures is Rs.70,

the cost of carry gain is Rs.10, but

the loss on stock is Rs.40). As more

and more dealers indulge in this, the

Futures price adjusts itself to the

theoretically correct level.

Two aspects need elaboration here.

When we say, “buy spot” or “sell

spot”, we refer to the buying or

selling of all the stocks that

constitute the Index in the same proportion as in the Index. This may, at

first sight, seem an impossible task given the fractional composition of the

stocks in the Index. Computer programs enable fast and accurate calculation

of the required quantities.

The second aspect to be noted is that a “buy” of Futures when carried to the

expiry does not entitle the buyer to delivery of the underlying stock.

Actually, the Futures contracts do not result in a delivery situation at all, If

not squared-up on the last date of trading at the converged price, they will be

TODAY

Sale in spot Rs. 1250 (+)

Buy Futures Rs. 1220 (-)

AFTER ONE MONTH

Buy spot Rs. 1290 (-)

Sell Futures Rs. 1290 (+)

Gain in cost of carry Rs. 10

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deemed to have been squared up at the converged price on the last day. But

this does not alter the arbitrage situation at all. It would still be possible to

buy the stock on the spot market while squaring up the Futures contract.

Since the principle of convergence ensures that the Futures will be priced

exactly equal to the stock on the expiry, it does not make any difference

whether one does the opposite transaction in the spot market or the Futures

market.

Now, why should the actual Futures price vary from the theoretical Futures

price at all? The explanation is that arbitrage though theoretically available,

will not always be practicable. Transaction costs, absence of sellable stock,

difficulty in borrowing, or the absence of a suitable counter-party to conclude

deals, could contribute to this phenomenon.

Another aspect to be noted in this context is the risk-free interest rate needed

for arriving at the theoretical Futures price. For the theory to be absolutely

correct, money must be freely borrowable and lendable at the risk-free rate.

On account of transaction difficulties, the cost of carry might be reckoning a

slightly higher level of interest than the conceptual risk-free rate.

Dividend yield has been ignored in the above calculations. To the extent of

the dividend yield, the cost of carry will be lower. This is so because the

holder of the spot is partly compensated for his invested capital by the

dividend that he receives.

5.6 Beta and the Optimal Hedge Ratio Hedging through Stock Futures involves use of the market portfolio. The

Index is considered as the true representative of the market portfolio and has

a Beta of 1. All other portfolios have Betas in relation to this market

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portfolio. Betas are calculated by taking the covariance of individual

securities/portfolios with the market returns and establishing a relationship.

The concept of Beta is controversial as the calculations may differ with the

period reckoned and with what is considered as the market portfolio.

The Optimal Hedge ratio tells us the number of Futures contracts to be used

for hedging given a value of the portfolio now and the value of a typical

Futures contract.

If an asset manager has a portfolio worth Rs.15 lakh consisting of shares in

the same proportion as the Index itself, and if the size of a typical Futures

contract as specified by the exchange is Rs.3 lakh, then she will have to go in

for 5 Futures (Rs.15 lakh portfolio value divided by Rs.3 lakh being the value

of one contract) for hedging purposes. In this case the calculation is

straightforward as the asset manager’s holding is a market portfolio and

therefore has a Beta of 1.

If the portfolio held is different from the market portfolio and has a Beta of

say 0.8, then the value to be hedged using the Index Futures is 0.8 multiplied

by Rs.15 Lakh = Rs.12 lakh and since one contract is of Rs.3 lakh, 4 contracts

will suffice. The reason for the difference between the two situations is that

in the former the Beta being 1, any change in the market portfolio is fully

represented in the Index. In the latter case, the portfolio is different and

experience tells us that we require only lesser number of Futures contracts to

hedge our position, since the volatility of the portfolio is less than that of the

Index. Conversely if the portfolio Beta had been 1.2, the value to be hedged

would have been 1.2 multiplied by Rs.15 lakh, coming to Rs.18 lakh, divided

by the standard contract size of Rs.3 lakh, = 6 contracts.

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5.7 Increasing and Decreasing Beta Stock indices can be used for temporarily increasing the Beta of a portfolio.

Let us take the case of an asset manager who has a Rs.15 Lakh portfolio with

a Beta of 0.8. She feels that the market is showing a bullish trend and in

order to participate in the upward movement wants to temporarily increase

the Beta of her portfolio to say 1.2. The most obvious way of achieving this

would be to buy new stocks with a higher level of Beta and simultaneously

selling some stock from the existing portfolio in such a way that the Beta is

on a weighted-average basis 1.2. However, this will involve complications of

rebalancing the portfolio and consequent transaction expenses. Besides her

intention is not to permanently change the portfolio but only to change it

temporarily. The answer for her would be to go in for Index Futures to the

extent of the difference in the two Betas.

She will buy Index Futures to the extent of -> {Portfolio Value now * (Desired

Beta – Existing Beta)/ Value of 1 contract}. In the above example her target

Beta is 1.2 as against the existing Beta of 0.8. The difference is 0.4 to which

extent she wants new contracts. Thus she has to take Futures to the extent of

Rs.6 Lakh ( 15*(1.2-0.8)). Since each contract is for Rs.3 lakh, she has to buy

2 contracts.

If the market does go up as she expects, she will have a gain from the

Futures position apart from whatever gains she has from the portfolio itself.

In the above example if she had feared a temporary fall in the market and

would have liked to reduce her Beta to say 0.6, she would have shorted

Futures as follows.

The portfolio value is Rs.15 lakh and her present Beta is 0.8. She wants to

reduce it to 0.6. To the extent of Rs.15 lakh *(0.8-0.6) = 3 lakh, she will short

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Futures. This corresponds to 1 contract since the size of 1 contract is Rs.3

lakh. If as she expects the prices fall, she will gain from the Futures short

position.

The following points should be noted in their connection:

• At first glance this might look like pure speculation. But if done with a

proper estimation of Beta, this could bring in short-term profits.

• This strategy of increasing and decreasing Beta should be based on

covered position of stocks and based on safeguards against excessive

positions.

• The strategy of increasing and decreasing Beta is apart from whatever

hedging that the asset manager is already doing. This is in addition

and not a substitute for regular hedging

5.8 Other uses of stock Futures Apart from the above uses, stock Futures are used for the following

additional purposes:

• Stock Futures can be used for enhanced portfolio management. A long

position in a portfolio of stocks is identical in pay off to corresponding

Futures over the horizon of the Futures contract. Of course, stock

related income like stock-lending and dividends are not there in the

Futures. But Futures require much less capital investment. So a

portfolio manager can use Stock Futures judiciously, in combination

with risk-free investments to achieve the same payoff as the Stock

portfolio itself. Of course, this should not be totally speculative and

ought to be based on judicious norms of portfolio management

• Sometimes stock Futures are used to take advantage of specific

unsystematic risk. For instance if a portfolio consists of stocks with

substantial unsystematic risk which is expected to become favorable in

the short- run. However, a general fall in the markets is expected and

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this will offset the gains from the specific security with unsystematic

risk. To continue gaining on this, the portfolio manager will short

Stock Futures to the extent of the decline expected and make gains

there from. Of course this will be apart from whatever hedging she

will carry out for the general decline in the market.

• For selective risk management of portfolios. This will involve a

strategy for participating in market movements through Futures while

being aggressive with the rest of the portfolio.

It should be noted that some of the higher uses of Futures look very much

like speculation, but is best indulged in based on specific strategies.

Portfolio managers having a horizon greater than 3 months, use the

technique of rolling the hedge forward to keep their hedges intact. Of course,

tracking errors and transaction costs make these difficult.

Dealers in the stock market also look for Calendar Spreads. If the cost of

carry principle is not uniformly held as between the Futures of various

months a strategy of selling the “overpriced” Futures and buying the “under

priced” Futures can be embarked upon, with the belief that the market will

correct itself and restore the cost of carry uniformly. The temporary anomaly

can be exploited for quick gains. For instance, if the 2-month Futures is

disproportionately high compared to the Spot and the 1-month Futures, the

2-month Futures can be sold and simultaneously the 1-month Futures can be

bought. If the market corrects the discrepancy, the two Futures will have the

correct difference between them. If this correction takes place within 1

month, the original positions can be reversed for profits.


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5.9 Illustrations Table I.5.1 Illustration showing convergence of

Futures prices to Spot prices in respect of Bajaj Auto

(Amount in Rs.)

Date Futures Settle Price Spot Price

18-Nov-05 1991.9 2,001.00 21-Nov-05 2038.05 2,049.80 22-Nov-05 2045.05 2,054.10 23-Nov-05 2066.4 2,064.60 24-Nov-05 2078.1 2,066.00 25-Nov-05 2086.6 2,064.95 26-Nov-05 2085.15 2,063.90 28-Nov-05 2138.4 2,117.15 29-Nov-05 2091.6 2,071.15 30-Nov-05 2024.35 2,009.95 1-Dec-05 2037.6 2,018.80 2-Dec-05 1998.7 1,983.30 5-Dec-05 2015.85 2,000.10 6-Dec-05 1996.75 1,986.70 7-Dec-05 2005.45 1,990.60 8-Dec-05 2027.8 2,015.25 9-Dec-05 2094.7 2,081.80 12-Dec-05 2128.45 2,116.30 13-Dec-05 2134.05 2,113.70 14-Dec-05 2132.15 2,117.45 15-Dec-05 2094.8 2,081.35 16-Dec-05 2132.55 2,118.05 19-Dec-05 2121.2 2,105.45 20-Dec-05 2095.9 2,082.90 21-Dec-05 2100.7 2,088.35 22-Dec-05 2113.2 2,099.00 23-Dec-05 2085.1 2,080.50 26-Dec-05 2028.75 2,020.45 27-Dec-05 2050.55 2,042.65 28-Dec-05 2002.85 1,998.25 29-Dec-05 1999.85 1,999.85


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5.10 Summary Stock Futures constitute an important derivative and one that is most

popular. These Futures have a number of advanced and straight applications

Stock indices are based on the principle of market portfolio. A market

portfolio is a typical selection of stocks in the market which represent the

market fairly accurately.

The Index is carefully chosen based on market-weights and certain other

criteria for liquidity.

Based on the estimation of the Beta of a portfolio and its relationship to the

market, it is possible to evolve several strategies using Index Futures. These

involve basis hedging, working on increasing the Beta and reducing the Beta

and some other sophisticated applications

While some of the strategies of an asset manager using Index Futures look

like speculation, they are based on principles of strategy and will be

successful in the long run only if based on a plan.

Index Futures are rampantly used for hedging. Hedging follows the same

principles as the regular asset-hedging, but the calculation of the Optimal

Hedge Ratio assumes importance here. This is based on the respective Betas

of the portfolio as compared to the market and suitably adjusted to come to

the market lot.

Traders use Calendar Spreads to take advantage of the discrepancy in the

prices among various Futures on the same underlying stock. This is done

with the hope the market will correct the discrepancy within the life of the

Futures.

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5.11 Key words

• Beta

• Optimal Hedge Ratio

• Portfolio Value

• Enhanced Portfolio management

• Rolling the hedge Forward

• Calendar Spreads

5.12 Questions for Self- study 1) What are the principal differences in Stock Futures as compared to

other types of Futures?

2) Does the strategy of increasing Beta work independently of hedging

practices?

3) Does the cost of carry principle work correctly with Stock Futures?

4) What is convergence? Why does it occur?


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Module 2 –

INTRODUCTION TO OPTIONS


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1 Types of Options


• To introduce the instrument of Options

• To understand the basic differences of Options from Forwards and

Futures

• To understand the payoff of various players in the Options market.

1.2 Introduction The subject of Options has been familiar to traders and investors in many

ways. Whenever we get an opportunity to delay a decision without the

foregoing of any rights it is an option. Thus the offer coupon that comes

along with the daily newspaper inviting us to shop in a particular mall and

get a discount of 10% on purchases by producing the coupon is an example of

an option. The buyer of the option – in this case the newspaper reader- is

under no obligation to go to the mall and buy. Yet, he has a right to shop to

get a discount. It is up to him to exercise the right or not according to his

convenience.

In the same way, sometimes vendors of goods give buyers a choice of sell-back

within a particular time and at a particular price. Thus if the buyer is not

happy with the performance of the product he can return it and obtain the

amount pre-determined. Here again, the customer is under no obligation to

sell, but can sell and obtain the price if he wants.


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The above are just two examples of a variety of Options we enjoy and give in

our regular lives. What makes the study of Options exciting is the fact that

there are multiple possibilities and strategies that arise out of these

instruments. Companies have discovered in recent years the great advantage

of these instruments for their risk management practices.

Options have also attracted a lot of criticism for their abject misuse by

certain operators. The example of Barings disaster is often quoted for

establishing the dangers of this instrument. It should be noted that misuse

of a good thing by certain elements should not result in the instrument,

which is otherwise useful being dismissed as worthless. Necessary

regulatory measures will result in greater control over misuse, while

retaining all the advantages.

1.3 Types of Options and option terminology Let us look at common Options terminology:

Calls Calls are rights without obligation to buy a certain underlying after a certain

period at a specified price. The buyer of the call gets a right to buy the

underlying asset, without any obligation to do so. He can enforce his right

after the specified time, if conditions favor such an action. Or, if conditions

are not favorable he can discard the right. All he will lose is what he has

paid originally as the price for the right.

Here, the enforcement of his right is called “exercise”. Any exercise has to be

at the pre-determined price, called the Exercise Price or Strike Price.


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The price the buyer pays for getting the right is called “premium”. As will be

seen in subsequent Chapters, a number of considerations come into play in

the determination of this price.

A seller of a call is also called the “writer” of the call

The buyer of the call will enforce his right if the price of the underlying falls

above the Exercise Price. If the price is below the Exercise Price he will

discard the call.

Puts

Puts are rights without obligation to sell a certain underlying asset after a

particular period at a specified price. The buyer of the put gets a right to sell

the underlying asset, without any obligation to do so. He can enforce his

right after the specified time, if conditions favor such an action. Or, if

conditions are not favorable he can discard the right. All he will lose is what

he has paid originally as the price for the right. As in a call, the enforcement

of his right is called “exercise”. Any exercise has to be at the pre-determined

price, called the Exercise Price or Strike Price.

The price the buyer pays for getting the right is called “premium”. As will be

seen in subsequent Chapters, a number of considerations play a role in the

determination of this price.

A seller of a put is also called the “writer” of the put

It can be safely concluded that the buyer of the put will enforce his right if

the price of the underlying falls below the Exercise Price. If the price is above

the Exercise Price he will discard the put.


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American and European Options

If the option is exercisable only after the expiry of a particular period, it is

called a European Option. Thus if A buys a right to buy an asset from B after

3 months, it is a European Option. However, if A has the right to buy the

asset at any time during the 3 months, it is called an American option. An

American Option can be exercised at any time during the tenure of the

contract. The word “European” and “American” has nothing to do with any

geographic location and is just a matter of terminology.

At-the money, Out-of the money, In-the-money

A call is said to be at the money when the actual asset price is around the

level of the strike price. If the asset value exceeds the strike price then the

call is in the money. On the other hand, if the asset price is less than the

Strike price, then the call is out of the money. The converse of the above

applies to puts. An option that is very much in the money is also called “deep

in the money”, and an option that is very much out of the money is also called

“ deep out of the money”

Naked Options and Covered Options

When an option position is taken up by a dealer without any position in the

underlying, it is called a naked position. Naked positions are speculative in

nature and are not entered into for risk management. A covered position, on

the other hand, signifies that the dealer has a position in the underlying

asset and is possibly hedging this. A great deal of the positive uses of

Options stems from covered positions. In recent years, many operators have

commenced having Option combinations that result in mimicking covered

positions.


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1.4 The question of exercise In a European Option a question of exercise arises only at the expiry of the

time period agreed upon. The buyer of the call will exercise if the underlying

asset price is greater than the Exercise Price. As we have seen he will

discard the call if the price happens to be less than the Exercise Price. The

theory of Options is based upon rational individuals and hence it is implicitly

assumed that when an European call expires in the money, it will be

exercised.

American calls can be exercised at any time before the expiry. Hence the

buyer strictly need not wait till the expiry for all exercise if the price of the

underline asset exceeds the strike price at any time during the tenure, he is

within his rights to exercise. However, as we will see in a later unit it can be

shown that it is not optimal for a buyer to exercise the American power ahead

of maturity.

A European put will be exercised by the buyer on expiry if the underlying

asset price is less than the Exercise Price. Here again the theory of Options

assumes investors to be rational and assumes that any put that expires in

the money will be exercised. An American put can be exercised at any time

during the tenure of the contract. Unlike the American call, it can be shown

that it is optimal to exercise the American put ahead of the expiry if the put

is sufficiently in the money.

In exchange-traded Options the European Options buyer does not have to

specifically indicate that he is exercising the Options. If, on expiry of the

contract, the underlying asset price is greater than the strike price the

difference will be treated as profits of the buyer by the exchange directly. In


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American Options, however, the buyer will have to specifically indicate that

he is exercising for the exchange to take note of this.

1.5 Options markets Options are traded in stock exchanges in a manner similar to the trading of

Futures contract. At any given point of time there will be three outstanding

Options. One option will expire in the current month, a second option in the

subsequent month and the third option two months hence. Trading is carried

out in various Exercise Prices. Whenever the underlying asset prices change

trading is allowed on new strike price to correspond to the new levels of the

underlying asset price. As with the parties to a Futures transaction, writers

of Options are required to pay margins. It can be observed that buyers of

Options need not pay any margin since they are not exposed to any risk.

A large segment of the Options market is speculative and the dealers have

naked positions in the Options and either buy or write only for speculative

purposes. Stringent margins are imposed by the exchange to avoid huge

losses and consequent pay out problems.

It has been observed that in the initial years of the introduction of Options

the premia charged for calls and puts are very high. This is particularly

because the writers are over –cautious about their exposure and would like to

be amply compensated. As markets mature option combinations by various

dealers result in more rational prices.

Options are traded in the National Stock Exchange (NSE). Details

regarding, trading and settlement can be had from their website www.nse-

india.com. The key features are:


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Options contracts in NIFTY have a maximum of 3-month trading cycle - the

near month (one), the next month (two) and the far month (three).

The contracts expire on the last Thursday of the expiry month. The Exchange

provides five strike prices for every option type (i.e. call & put) during the

trading month. At any time, there are two contracts in-the-money, two

contracts out-of-the-money and one contract at-the-money. The strike price

interval is Rs.10. The permitted lot size of S&P CNX Nifty Options contracts

is 200 and multiples thereof.

In addition, NSE also permits Options on individual securities. These

Options are American, while the NIFTY Options are European. The trading

cycle is the same as the NIFTY Options, but some changes in the lot sizes are

made from time to time.

1.6 Differences between Options and Futures • A long Futures contract obliges the buyer to buy the underlying on

expiry of the contract at the pre-determined price. A long call enables

the buyer to buy the underlying if circumstances so warrant . There is

no compulsion to purchase unlike the Futures contract.

• A call option entails payment of premium by the buyer to the seller. A

Futures contract does not attract any premium.

• The seller of a Futures contract is obliged to sell the asset at the price

stipulated. The writer of the call is similarly obliged to sell the asset at

the price stipulated. The only difference is that the writer of the call

gets a premium while the seller of the Futures does not.

• Selling Futures is comparable to buying a put. However, the buying of

a put results in payment of a premium.


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• Both the buyer and the seller of the Futures contract have to pay

various types of margin to the exchange. Only the seller of the option

need pay margin.

• Even if the final price of the underlying asset ends up lower than

expected, a buyer of a Futures contract still has to buy the asset at the

agreed price. A buyer of a call is under no such obligation.

1.7 Summary Options are a part of our everyday transaction. Broadly, Options refer to a

right without an obligation to buy or sell an asset during or after a particular

time at a specified price. A direct deal in the underlying asset involves

buying or selling the asset itself. In an Options contract the buying or selling

is not of the asset but of a right to buy or sell the asset.

A call Option signifies the right to buy while a put Option signifies a right to

sell. The seller of a call or a put is called a Writer of the Options. The

enforcement of a right to buy or sell is called “exercise”

A buyer of a call will exercise his right if the underlying asset price exceeds

the strike price. A buyer of a put will have this right if the underlying asset

price is below the strike price.

Options are of two types- European and American. European Options can be

exercised only on the expiry of the tenure of the contract, while an American

Option can be exercised at any time during the tenure of the contract.

An Option is said to be” in the money” if the asset price is greater than the

Exercise Price for a call: or if the asset price is less than the Exercise Price for

a put. An Option is said to be” out of the money” if the asset price is less than


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the Exercise Price for a call: or if the asset price is greater than the exercise

for a put.

1.8 Key words

• Call

• Put

• Writer of an Option

• European Option

• American Option

• At the Money

• In the Money

• Out of the Money

• Strike Price

1.9 Questions for Self- study 1) How are Options different from Futures?

2) Is buying the call same as writing a put?

3) What are the margin requirements generally imposed by the

exchanges for Options?

4) When is a put exercised?


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2 Pay off of various Options


• To draw up the payoff of the buyers and sellers of calls

• To draw up the payoff of the buyers and sellers of puts

• To understand from the patterns of payoff the broad strategies

investors follow for various Options-related gains

2.2 Introduction In the last unit we have seen the basic definitions of various Options and

their general pattern of enforcement. The potential profits and maximum

losses are different for the various players in the Options market. We now

look at the general payoff of the buyers and sellers separately and draw

conclusions as to their motivations in entering into these positions.

The examples in this unit are based on stock prices. This is because it is

intuitively easier to understand the instrument of Options through stocks

more than any other asset. Stocks are generally well traded and their price

movements are very transparent. The Options market in respect of stocks

also gets good coverage in newspapers and business dailies and therefore the

investors are well-versed with price movements. However, the broad

principles of payoff will apply for any other asset as well.

Stock exchanges impose margins on sellers of Options for their positions.

These margins could be in the nature of Initial Margin, Maintenance Margin

and Mark-to-Market margins. As we will soon see, sellers are exposed to

considerable risk in the Option segment and sometimes these can result in


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total disaster. It is, therefore, imperative that the payoff profile is well

understood before a trader embarks upon a selling position in Options.

2.3 Payoff of long and short call

Buying a call Table II.2.1 – Payoff from buying a call

Asset price today 120

Strike price 125

Call premium 4

BUYING A CALL

If end asset price is

Gain from exercising

call

Premium paid

Net gain

120 0 -4 -4

121 0 -4 -4

122 0 -4 -4

123 0 -4 -4

124 0 -4 -4

125 0 -4 -4

126 1 -4 -3

127 2 -4 -2

128 3 -4 -1

129 4 -4 0

130 5 -4 1

131 6 -4 2

132 7 -4 3

133 8 -4 4

134 9 -4 5

135 10 -4 6

136 11 -4 7

137 12 -4 8

138 13 -4 9

139 14 -4 10

140 15 -4 11


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The buyer of a call will gain whenever the price of the asset exceeds the

strike price. However, since he has to pay the premium the gains will arise

only after the premium amount has been recovered beyond the Exercise

Price. If the asset price ends up below the Exercise Price, the buyer will

discard the call, resulting in a loss of the premium amount paid. The buyer of

an American call can exercise this right at any time during the tenure of the

contract.

Thus the maximum loss that he can incur is only the premium amount, while

the maximum profit can be infinite.

Selling a call The seller of a call will gain whenever the price of the asset ends up below the

strike price. The initial premium gained by him will be his total gain in such

circumstances. However, if the price of the asset exceeds the strike price, he

will suffer losses. The premium initially received will offset the losses for a

small extent, but as the price ends up higher, his losses are greater. The

seller of an American call is exposed to this risk at any time during the

tenure of the contract.

Thus the maximum loss that he can incur is infinite, while the maximum

profit will be the premium received.


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Table II.2.2 Payoff from selling a call Asset price today 120

Strike price 125

Call premium 4

WRITING A CALL


Loss from call being exercised

Premium received

Net gain

121 0 4 4

122 0 4 4

123 0 4 4

124 0 4 4

125 0 4 4

126 -1 4 3

127 -2 4 2

128 -3 4 1

129 -4 4 0

130 -5 4 -1

131 -6 4 -2

132 -7 4 -3

133 -8 4 -4

134 -9 4 -5

135 -10 4 -6

136 -11 4 -7

137 -12 4 -8

138 -13 4 -9

139 -14 4 -10

140 -15 4 -11


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2.4 Payoff of long and short put

Buying a put Table II.2.3 Payoff from buying a put


Strike price 125

Put premium 4

BUYING A PUT


Gain from exercising

put

Premium paid

Net gain

117 8 -4 4

118 7 -4 3

119 6 -4 2

120 5 -4 1

121 4 -4 0

122 3 -4 -1

123 2 -4 -2

124 1 -4 -3

125 0 -4 -4

126 0 -4 -4

127 0 -4 -4

128 0 -4 -4

129 0 -4 -4

130 0 -4 -4

131 0 -4 -4

132 0 -4 -4

133 0 -4 -4

134 0 -4 -4

135 0 -4 -4

136 0 -4 -4


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The buyer of a put will gain whenever the price of the asset ends up below

the strike price. However, since he has to pay the premium the gains will

arise only after the premium amount has been recovered. If the asset price

ends up above the Exercise Price, the buyer will discard the put, resulting in

a loss of the premium amount paid. The buyer of an American put can

exercise this right at any time during the tenure of the contract.

Thus the maximum loss that he can incur is only the premium amount, while

the maximum profit can be the strike price. The asset value cannot be

negative and the lowest it can reach is only 0. In such an eventuality, he will

get a gain of (Strike price -0.). Hence, the maximum gain that he can get is

the Strike Price.

Selling a put The seller of a put will gain whenever the price of the asset ends up above the

strike price. The initial premium gained by him will be his total gain in such

circumstances. However, if the price of the asset falls below the strike price,

he will suffer losses. The premium initially received will offset the losses for

a small extent, but as the price ends up lower, his losses are greater. The

seller of an American call is exposed to this risk at any time during the

tenure of the contract.

Thus the maximum loss that he can incur is the Strike Price (since the asset

value cannot go below 0), while the maximum profit will be the premium

received.


85

Table II.2.4 Payoff from selling a put


Strike price 125

Put premium 4

SELLING A PUT


Loss from put being exercised

Premium received

Net gain

117 -8 4 -4

118 -7 4 -3

119 -6 4 -2

120 -5 4 -1

121 -4 4 0

122 -3 4 1

123 -2 4 2

124 -1 4 3

125 0 4 4

126 0 4 4

127 0 4 4

128 0 4 4

129 0 4 4

130 0 4 4

131 0 4 4

132 0 4 4

133 0 4 4

134 0 4 4

135 0 4 4

136 0 4 4


86

2.5 Risk and premium The possible motivations of the various players in the Options market are

examined below:

1. The buyer of a call does so with the desire to participate in an upward

movement of the asset prices. The alternative would have been to buy

the asset itself. But that would be risky in the sense that if the value

comes down, he would suffer portfolio losses. In order to have the best

of both worlds (to gain if the prices go up and not to lose more than the

premium if the prices come down), a call is bought. The call that is

bought comes at a premium. The premium is determined partly by

demand and supply and partly by intrinsic worth of the call. In the

initial years of the introduction of Options in any exchange, call premia

are observed to be disproportionately high and there are few sellers in

the market. As markets mature this changes and the prices come to

their intrinsic worth. Pricing is a major discussion as far as Options

are concerned and this gets covered in a subsequent unit.

2. The buyer of a put expects the market to take a downward slide, but is

not fully confident about it. The alternative would be to short sell the

asset, but that would be dangerous, if the prices go up contrary to

expectations. The buyer of a put also has the best of both worlds (he

participates in profits if the prices do come down, and he only loses the

premium if the prices go up). The bought put comes at a premium.

Like call premia, put premia is also decided by market forces and the

intrinsic value of the put. The pricing of puts involves certain no-

arbitrage conditions, which get covered in a later unit.

3. The seller of a call expects the price to remain steady, or go down and

in any case does not expect the price to rise radically. Some sellers of

calls are speculators who do so for making quick profits. Buyers of


87

calls feel that the premia paid by them for getting this choice is trivial.

But as several buyers enter into such transactions, the premia gain

becomes substantial for the sellers. Occasionally, sellers have a

covered position in the assets. This is a strategy called covered call

writing, which will be discussed in a later unit.

4. The seller of a put expects the price to remain steady or go up, and

does not expect the prices to fall. Sellers of puts are sometimes

speculators who seek to make gains out of the premia collected.

However, the risks are high and sometimes they end up suffering huge

losses. A put can also be written with covered position in stocks for

strategic reasons.

2.6 Illustrations Table II.2.5 Illustration on payoff of various Options

Payoff Calculation – Stock prices in Rs..

Given Pay off Calculation

Stock price on

Jan 1, 2007

Exercise Price

Put option

premium as on Jan 1

Call option

premium as on Jan 1

Stock price on expiry of

call

Call, Strike price = 55

Call, Strike price = 60

Put, Strike price = 55

Put, Strike price = 60

55 55 2.625 2.875 52 -2.875 -1.75 0.375 2.5

55 60 5.5 1.75 53 -2.875 -1.75 -0.625 1.5

54 -2.875 -1.75 -1.625 0.5

55 -2.875 -1.75 -2.625 -0.5

56 -1.875 -1.75 -2.625 -1.5

57 -0.875 -1.75 -2.625 -2.5

58 0.125 -1.75 -2.625 -3.5

59 1.125 -1.75 -2.625 -4.5

60 2.125 -1.75 -2.625 -5.5

61 3.125 -0.75 -2.625 -5.5

62 4.125 0.25 -2.625 -5.5

63 5.125 1.25 -2.625 -5.5


88

Pay off for Call at strike = 55

-4

-2

0

2

4

6

8

52 53 54 55 56 57 58 59 60 61 62 63 64 65

Stock Price

Pay off

Pay off for Call at strike = 60

-2

-1

0

1

2

3

4

52 53 54 55 56 57 58 59 60 61 62 63 64 65

Stock Price

Pay off


89

2.7 Summary The payoff of various players in the Options market depends upon the

possible range of final prices and the premium paid/received initially. A

buyer of a call gains if the asset price exceeds the strike price and does not

lose when the asset price ends up below the strike price. The buyer of the call

has a maximum loss of the premium and an infinite potential for gains.

Pay off for Put at strike = 55

-3 -2.5

-2 -1.5

-1 -0.5

0

0.5 1

52 53 54 55 56 57 58 59 60 61 62 63 64 65

Stock Price

Pay off

Pay off for Put at strike = 60

-6 -5 -4 -3 -2 -1 0

1

2

3

52 53 54 55 56 57 58 59 60 61 62 63 64 65

Stock Price

Pay off

Series1


90

A buyer of a put has a maximum gain of the Strike Price and a maximum loss

of the premium. The put buyer will gain whenever the asset ends up below

the strike price

The seller of a call earns a premium upfront but is exposed to the risk of the

asset price ending up higher than the strike price. His potential losses are

infinite, while his maximum gain is the premium collected

The seller of a put earns a premium upfront but is exposed to the risk of the

asset price ending up lower than the strike price. His potential losses are up

to the Strike Price, while his maximum gain is the premium collected

2.8 Key words

• Payoff

• Speculative position

• Covered position

2.9 Questions for Self- study 1) What is the maximum pay off of a buyer of a put? Why is it not

infinite like that of the buyer of the call?

2) Does the seller of a call take greater risk than the seller of a put?

3) Is buying a put the same as selling a call?

4) Who takes the greater risk – the seller of a put or the buyer of a call?


91

3 Special applications of Options


• To look at common strategies using Options

• To understand the payoff under these strategies

• To look at the general concept of synthetic portfolios

• To get an introduction to the use of synthetic portfolios in valuation

3.2 Introduction As seen in the previous unit, the payoff situation is different for the buyer

and the seller. Option positions can be naked or covered. Naked positions are

speculative in nature and do not conform to any analytical theory. We are

therefore concerned basically with covered positions and combinations, which

can stem out of a strategy.

The two most common strategies followed by dealers in Options are the

Covered Call Strategy and the Protective Put Strategy. These are explained

in detail hereunder.

3.3 Covered Call writing In a covered call writing, the investor will have the holding of a portfolio or a

Company share, and will write calls on the same share/s.

For instance, an investor holding 2000 shares of Punjab Tractor can write

(sell) calls for 2000 shares. In the process, a premium is earned on the calls

sold

1. If on expiry, the stock price exceeds the Exercise Price, the call will be

exercised against the investor. Since theoretically, there is no upper


92

limit to the stock prices on expiry, the investor is actually open to

unlimited loss. The loss is, however, cushioned to the extent of the

premium earned on selling the calls.

2. If the stock price ends up equal to or less than the Exercise Price, it

will not be exercised against the investor. So the initial gain from

selling the call will be the net gain from the transaction.

3. The underlying value of the portfolio also goes up with any rise in the

prices. So the loss on the call writing is compensated by the rise in

portfolio value in equal measure. Effectively, it comes down to having

sold the portfolio at the Exercise Price.

Let us take an illustration:

In the above case, assume that Punjab Tractor is currently going at Rs.320.

Let us say our investor has written calls at an Exercise Price of Rs.320. If

Punjab Tractor ends up at Rs.360 on expiry, there will be a loss of Rs.40 (360-

320). This is the similar to selling the shares at Rs.320, which is the Exercise

Price.

Let us now look at situations that will be ideal for entering into covered call

writing.

a) When the market is listless and does not appear to be likely to have big

movements in the short-run, the portfolio owner could write covered

calls. By doing so, the portfolio manager is seeking to enhance the

value of the portfolio by earning extra income on the call writing.

When the call is written for a short tenure, the volatility is unlikely to

be big, and the opportunity loss is also not likely to be high.


93

b) A portfolio manager, who has assessed that the overall variation in the

markets is not likely to be substantial, can keep on playing this

strategy over and over again. While there would be occasional losses,

the strategy is likely to return steady income over a long duration.

This would work particularly if the manager is a “buy and hold”

investor and does not intend to sell in the near future.

Two other strategic considerations come into play in respect of covered call

writing. The writer has the choice of the Exercise Price. Naturally, higher

the Exercise Price, the lower the premium he can charge, but correspondingly

the risk level changes too. In order to strike the right balance between the

premium that can be earned and the risk that is borne in the process, the

possible outcomes and returns can be drawn up for various scenarios and the

ideal level arrived at.

Secondly, the writer has the choice of reversing the position prior to expiry

should the situation begin to look dangerous. Alternatively, he can soften

losses by buying calls.

Some studies have shown covered call writing to be more profitable than

buying naked calls outright. However, if this is really so at all times, then

there will be a scramble for writing calls and that process itself will reduce

the attractiveness of the premium. In the initial stages of derivative market

acceptance, it is likely that a number of new players like the conceptual “win-

win” of buying a call and would therefore provide a good market for the

covered call writing strategy. Professional fund managers, can, in such a

situation reap rich rewards by having a portfolio of covered calls.

An example of the full payoff is given below:


94

Table II.3.1 Payoff from a Covered Call Writing strategy

(Amount in Rs.)

Asset value 100

Call written for 4

Ex. price of call written 102

At expiry price Portfolio value

Net Gain/loss from call writing

Net value at end

95 95 4 99

96 96 4 100

97 97 4 101

98 98 4 102

99 99 4 103

100 100 4 104

101 101 4 105

102 102 4 106

103 103 3 106

104 104 2 106

105 105 1 106

106 106 0 106

107 107 -1 106

108 108 -2 106

109 109 -3 106

110 110 -4 106

111 111 -5 106

112 112 -6 106

Another example is shown below. The Asset Value here is Rs.55 and the Call

premium is Rs.2.875 for an Exercise Price of Rs.55. The example is continued

with a covered call on the same stock with an Exercise price of Rs.60, written

for Rs.1.75


95

Table II.3.2 Covered Call writing – illustration

(Amount in Rs.) Covered Call Strategy

Asset value 55

Call written for 2.875

Ex.price of call written 55



Net value at end

52 52 2.875 54.875

53 53 2.875 55.875

54 54 2.875 56.875

55 55 2.875 57.875

56 56 1.875 57.875

57 57 0.875 57.875

58 58 -0.125 57.875

59 59 -1.125 57.875

60 60 -2.125 57.875

61 61 -3.125 57.875

62 62 -4.125 57.875

63 63 -5.125 57.875

64 64 -6.125 57.875

65 65 -7.125 57.875


96

Covered call with strike 55

5353.5

5454.5

5555.5

5656.5

5757.5

5858.5

52 53 54 55 56 57 58 59 60 61 62 63 64 65

Stock price at expiry

Valu

e of

por

tfol

io $

Table II.3.3 Covered call writing- Exercise Price Rs.60

(Amount in Rs.)

Asset value 55

Call written for 1.75




Net value at end

54 54 1.75 55.75

55 55 1.75 56.75

56 56 1.75 57.75

57 57 1.75 58.75

58 58 1.75 59.75

59 59 1.75 60.75

60 60 1.75 61.75

61 61 0.75 61.75

62 62 -0.25 61.75

63 63 -1.25 61.75

64 64 -2.25 61.75

65 65 -3.25 61.75


97

Covered call with strike 60

48

50

52

54

56

58

60

62

64

52 53 54 55 56 57 58 59 60 61 62 63 64 65

Stock price

Valu

e of

por

tfol

io $

3.4 Protective Put strategy A Protective Put strategy is similar to the selling of a Futures contract for

protecting a portfolio. When a portfolio manager fears that the value of her

portfolio might get eroded because of a fall in value, she may choose to protect

the portfolio by selling the appropriate number of Futures. The other choice

would have been to sell the stock itself. If the portfolio manager feels that

the fall is temporary and if she is happy with her portfolio composition, she

may choose to just sell the Futures for protection. Buying an appropriate

number of puts will serve the same purpose. If the asset value falls below the

strike level ( which is what the portfolio manager fears), the puts will give a

payoff of the difference between the strike price and the final price. Thus, if

the put is bought at around the level of the current portfolio value, any

depletion in value as a result of the fall in the prices will be compensated by

the put. This is called the Protective Put strategy.

There are important points of difference between the Protective Put strategy

and a strategy of selling Futures. A Protective Put gives protection without

committing the buyer to the strike price.


98

An example of Protective Puts is shown below:

Table II.3.4 – Protective Put Strategy

(Amount in Rs.)

Asset value 100

Call written for 4




Net value at end

95 95 1 96

96 96 0 96

97 97 -1 96

98 98 -2 96

99 99 -3 96

100 100 -4 96

101 101 -4 97

102 102 -4 98

103 103 -4 99

104 104 -4 100

105 105 -4 101

106 106 -4 102

107 107 -4 103

108 108 -4 104

109 109 -4 105

110 110 -4 106

111 111 -4 107

112 112 -4 108

Contrary to expectations, if the price of the portfolio goes up, the buyer of the

put can just discard the put and continue to enjoy any appreciation in the

portfolio. However, the seller of the Futures will be committed to the price

and will not be able to participate in the higher profits.


99

The Protective Put comes with a price in the form of a premium, while

Futures do not involve any premium.

3.5 Mimicking and synthetic portfolios A mimicking portfolio is one which has the same profit or loss as the original

portfolio, but not the same value. The mimicking portfolio will consist of

securities which are different from those in the main portfolio, but will yet

manage to give the same payoff.

The mimicking portfolio ends up with a profit or loss which is different from

the main portfolio by only Rs.4. Here the mimicking portfolio consisted of

buying a call and selling a put. The put that was sold fetched a premium,

while the call had to be bought at a premium. Thus there was a total cost of

Rs.4.

When the asset expires at various prices as shown in the table, there is a

payoff for the call and the put that has been written. The total position

corresponds to the main portfolio of the asset itself, subject to a difference of

Rs.4.


100

Table II.3.5 Mimicking portfolio (Amount in Rs.) Stock price 100

Call strike price 100

Put strike price 100

Buy call - pay 7

Sell put - get 3

Total cost 4

Stock at end Call value

Put value

Position cost

Total from mimic. port

Compare to loss in

portfolio 91 0 -9 -4 -13 -9

92 0 -8 -4 -12 -8

93 0 -7 -4 -11 -7

94 0 -6 -4 -10 -6

95 0 -5 -4 -9 -5

96 0 -4 -4 -8 -4

97 0 -3 -4 -7 -3

98 0 -2 -4 -6 -2

99 0 -1 -4 -5 -1

100 0 0 -4 -4 0

101 1 0 -4 -3 1

102 2 0 -4 -2 2

103 3 0 -4 -1 3

104 4 0 -4 0 4

105 5 0 -4 1 5

106 6 0 -4 2 6

107 7 0 -4 3 7

108 8 0 -4 4 8

109 9 0 -4 5 9

110 10 0 -4 6 10

In a synthetic portfolio, the value of the portfolio and its payoff both

correspond with the asset portfolio. In the above example, if a risk-free bond

costing Rs.96 and maturing at Rs.100 on expiry of the Options is also bought,


101

the total initial investment matches exactly with the portfolio and so does the

payoff.

Table II.3.6 Synthetic Portfolio (Amount in Rs.)

Stock price 100

Call strike price 100

Put strike price 100

Buy call – pay 7

Sell put – get 3

Total cost 4

Maturity value of discount bond 100

Cost of discount bond 96

Stock at end

Call value

Put value

Bond value

Value of synthetic portfolio

95 0 -5 100 95

96 0 -4 100 96

97 0 -3 100 97

98 0 -2 100 98

99 0 -1 100 99

100 0 0 100 100

101 1 0 100 101

102 2 0 100 102

103 3 0 100 103

104 4 0 100 104

105 5 0 100 105

106 6 0 100 106

107 7 0 100 107

108 8 0 100 108

109 9 0 100 109

110 10 0 100 110


102

3.6 Summary The basic pay off structure of Options gives rise to certain common strategies

that could be used for by portfolio managers. The most common of such

strategies are the Covered Call Strategy and the Protective Put Strategy.

In the Covered Call Strategy, the asset manager combines his portfolio with a

sold position in calls. The calls are written at such a level of the strike price

that the writer does not expect it to be exercised. Moreover, market

conditions may be such that price movements are within a narrow range.

However, if prices do shoot up and the call gets exercised, the asset manger

has the position in the stock to cover for the eventuality. The net result

would then be that the sale has been made at the strike price and a potential

for gains in the asset had been lost by the call having been written. Portfolio

managers use this strategy on a continuous basis and not for isolated

transactions.

The Protective Put strategy is an improvement on the short Futures contract

used by the asset managers to protect themselves against temporary fall in

asset values. The puts have to be bought at a premium and will enable the

asset manager to get protection in case the prices fall. If the prices do not

fall, the premium amount is the only loss.

Mimicking portfolios refer to a combination of Options with or without other

Derivatives and assets which result in almost identical payoff with the

parent asset, without involving the same initial investment.

A synthetic portfolio is like the mimicking portfolio, but has an initial

investment and final payoff both similar to the parent asset


103

3.7 Key words

• Covered call writing

• Protective Puts

• Mimicking portfolio

• Synthetic portfolio

3.8 Questions for Self- study 1) In what way does the Protective Put Strategy outperform the Short

Futures strategy?

2) “Covered Call writing does not work in a rising market”. Why?

3) How is the mimicking portfolio different from the synthetic portfolio?


104

4 Options bounds- Calls


• To understand the principles behind the upper and lower limits of

Options prices

• To understand the principle of arbitrage in price determination

• To understand the changes in these limits for American Options

4.2 Introduction As a first step to price determination, it is necessary to appreciate the

maximum and minimum limits to which Option prices can go. These limits

are based on principles which if violated, would create an opportunity for

risk-free arbitrage. Arbitrage can be performed by any player in the market

and with the force of more and more dealers doing it; prices will stabilize to

the correct levels.

Sometimes it is argued that the upper and lower limits are pure theoretical

values and do not have any practical relevance. It is said that prices will

never be near the limits as such and therefore the limits, by themselves do

not serve any purpose. However, this is a wrong notion. The limits to prices

reinforce our understanding regarding the economics behind various types of

Options, and also give a good picture of the factors governing prices.

We seek to do separate analysis in respect of calls and determine the no-

arbitrage maximum and minimum prices. We also then look at the principles

governing the determination of bounds of American call prices.


105

Options pricing consists of two elements – intrinsic value and time value.

Intrinsic value refers to the extent to which the Option is “in the money”, and

factors in interim dividends. The time value refers to the time available with

the buyer for exercising the option. Obviously, the more the time available,

the more valuable the option. The phenomenon of time value explains why

even Options which apparently do not have an intrinsic value still have

certain overall value.

4.3 Upper bounds of call prices

A European Call without dividends cannot have a value higher

than the current value of the underlying asset

This can be established with an example: Let us assume the stock price to be

Rs.40, and the call price to be Rs.41.( for a strike price of Rs.40)

Here, any dealer can write the call and buy the stock, for getting a difference

of Re.1. If on expiry the call is exercised against him, the stock position will

cover the deal. If the call goes unexercised, the stock can be sold in the

market for whatever small value and the dealer ends up getting a total of the

Rs.41 from writing the call and the sale proceeds of the stock. His potential

for profit is risk-free. Thus the call price can never exceed the value of the

asset

A European Call with dividends cannot have a value higher than

(the current value of the underlying asset minus expected

dividend)

The position does not change with availability of Dividends and so a

European call with dividends also cannot have a value greater than the asset

value. Further the value cannot even be higher than the value of the asset


106

less dividend. Otherwise, it will be possible to sell the call and buy the stock,

enjoy the dividends and use the stock to cover the call written.

In the above example if dividends are Rs.3 and are known with certainty, the

call value cannot exceed Rs.37 (40-3). If it exceeds Rs.37, the call can be sold

and the stock bought for Rs.37, pocketing the difference. A dividend will be

received of Rs.3 and with this the dealer has Rs.40 to cover for the call

written. If the stock price ends up higher than Rs.40, the call will be

exercised against the dealer, but he has the stock to cover this. If the stock

price ends up lower than Rs.40, the call will not be exercised against him,

and he can sell the stock for whatever price in the market for additional

gains.

4.4 Lower bounds of call prices

A European Call without dividends cannot have a value lower

than the difference between the Stock Price and the present value

of the Exercise Price.

This can be established with an example:

Suppose the Stock price is Rs.41 and the Strike Price is Rs.42. Discounting

the Strike Price at the risk-free interest, we get the Present Value of the

Strike Price to be Rs.40.50.


107

Table II.4.1 Lower bound of European Calls (no dividends)

(Amount in Rs.) Stock price 41

Strike price 42

Present value of strike price

40.5

The Present Value is calculated by discounting

the Strike price of Rs.42 at the risk-free rate.

Call cannot be less than

(Stock - PV of strike)

Cannot be lower than 0.5

if it is 0.25, say

Buy call spend 0.25

Sell stock get 41

Invest in risk-free

bonds 40.5

Immediate profit 0.25

At maturity

Get 42 from the bonds

Use it for exercising call to get

back shares

The gains are risk-free. Hence the bound of prices will be maintained.

A European Call with known dividends cannot have a value lower

than the (the Stock Price minus the present value of the Exercise

Price minus present value of dividend expected)

The following example will illustrate this (Amounts in Rs.)


108

Table II.4.2 Lower bound of European Calls (with dividends)

(Amount in Rs.) Stock 60

Strike 20

PV of strike 18.18

Time 1 year

Risk free rate 10%

PV of dividends 3

Lower bound is (60-18.18-3) 38.82

If call is going at say 38

Buy call spend 38

Sell stock get 60

invest spend18.18

immediate gain 3.82

at maturity get 20 from investment

Exercise call and get back stock

Loss on dividends 3

net gain 0.82

So the lower limit will have to be met.

4.5 Upper bounds of call prices-American Options

An American Call without dividends cannot have a value higher

than the current value of the underlying asset

Here the principle is the same as that of the European call. The fact that

exercise can be had at any time does not make a difference.


109

An American Call with dividends cannot have a value higher than

(the current value of the underlying asset minus expected

dividend)

The fact that exercise can be had at any time does not make a difference. The

upper bond is the same as that of a corresponding European Call.

4.6 Lower bounds of call prices-American Options

An American Call without dividends cannot have a value lower

than the difference between the Stock Price and the Exercise

Price.

Since the exercise of an American Option does not have to wait till maturity,

there is no need to take the Present Value of the Exercise Price for deciding

the lower bound.

This can be established by modifying our earlier example:

Table II.4.3. Lower bound of American call (no dividends)


Strike price 40

Call cannot be less than (Stock - Strike)

Cannot be lower than 1,

if it is 0.75, say

Buy call spend 0.75

Sell stock get 41

Exercise call Spend 40

Immediate profit 0.25


110

An American Call with known dividends cannot have a value

lower than the (the Stock Price minus the present value of the

Exercise Price minus present value of dividend expected)

The bound cannot be lower than the corresponding European call. An

example is shown to illustrate this:

Table II.4.4. Lower bound of American call (with dividends)


Strike price 115

Risk free interest 8%

Dividend expected 5

Time of dividend 1 month

Tenure 3 months

PV of ex. Price 112.7228

PV of dividend 4.966778

Lower Bound 4.310375

If it is say Rs.4

Buy call spend 4

Sell stock get 122

Invest in PV of Exercise Price pay 112.72

Invest in PV of dividend pay 4.96

Immediate gain 0.32

Now when the Investment on Exercise Price

matures, it can be used for exercising call

When the dividend investment matures the

Dividends can be taken.

4.7 Summary of principles of American Options pricing

• If the stock price is 0, the value of an American call must be 0.

• The minimum value of an American call is either 0, or the difference

between the Stock price and Strike Price, whichever is greater.


111

• An American call can never be worth more than the underlying asset

• For a stock that does not pay dividends than minimum value of a

European Option is 0, or the difference between the stock price and the

present value of the strike price whichever is greater.

• An American call can never be worth less than a European call.

• Two American calls on the same stock having the same Exercise Price

has to be priced such that the one with the longer maturity is worth as

much or more than the one with the shorter maturity.

• If the stock price is 0, the value of an American put must be its

Exercise Price.

• The minimum value of an American put is either 0 or the difference

between the Strike Price and Stock price at start, whichever is greater.

• The maximum value of an American put is its Exercise Price.

• An American put is worth at least as much as the European put.

4.8 Summary Call prices have to be within certain boundaries determined by no-arbitrage

conditions. These limits hold regardless of the specific values of the stock.

When it comes to American Options, the question of exercise involves a

number of considerations. While the bounds are not directly indicative of the

price of the call, they show the economics behind the working of the Options

market and tell us the factors that govern pricing.

4.9 Key words

• Bounds

• Early Exercise

• No-arbitrage condition

• Dividends


112

4.10 Questions for Self- study 1) Will there be circumstances when the upper limit of an American call

is different from the upper limit of a corresponding European call?

2) How does the factor of an interim dividend make a difference in the

lower bound of European calls?

3) What is the lower bound of an American Call without dividend?


113

5 Options bounds -Puts


• To understand the principles behind the upper and lower limits of

Options prices

• To understand the principle of arbitrage in price determination

• To understand the changes in these limits for American Options

5.2 Introduction In the previous unit we saw the upper and lower limits of European and

American calls with or without dividends. The no-arbitrage principles given

there will apply in equal measure to the determination of put bounds as well.

It should be remembered that that these bounds are not actually indicative of

the prices, but show only the maximum and minimum limits of these prices.

We seek to do separate analysis in respect of puts and determine the no-

arbitrage maximum and minimum prices. We also then look at the principles

governing the determination of bounds of American put prices.

It is worthwhile reiterating that Options pricing consists of two elements –

intrinsic value and time value. Intrinsic value refers to the extent to which

the Option is “in the money”, and factors in interim dividends. The time

value refers to the time available with the buyer for exercising the option.

Obviously, the more the time available the more valuable the option. The

phenomenon of time value explains why even Options which apparently do

not have an intrinsic value still have certain overall value.


114

5.3 Upper bounds of put prices

A European put without dividends cannot be greater than or

equal to the present value of the Strike price

Table II.5.1 Upper bound of European put (no dividends)


Strike price 45

PV of strike price 43

Price cannot be>

PV of strike

If price is 44

sell P get 44

Invest PV of K

invest

43

Worst case Put is exercised against the dealer

Loss 45

Dealer gets from inv. 45

Profit 1

A European put with dividends cannot have a value greater than

or equal to the present value of the Strike price

The position does not change with availability of Dividends and so a

European put with dividends also cannot have a value greater than the

present value of the Strike price.

Another principle here is that the maximum loss to be suffered by a put

writer will be only the strike price and if he is able to invest the present value


115

of that straightaway he is safe. The fact that there may be dividends on the

stock is not relevant because, the dealer is not looking at the stock at all, but

only at the strike price and the price of the put.

5.4 Lower bounds of put prices

A European put without dividends cannot have a price lower than

the difference between the present value of the Strike price and

the Stock price

The following illustration will establish this:

Table II.5.2. Lower bound of European put (no dividends)

(Amount in Rs.) Stock price

42

Strike price

45

PV of strike price

43

Must be greater than or equal to

Max of 0 or PV of strike - Stock

If price is 42

It cannot be lower than 1 (43-42)

if it is say 0.5

buy put spend 0.5

borrow 43

buy stock spend42

straight gain 0.5

At maturity, if Stock

exceeds Strike discard put

if stock is less than Strike, exercise put

Trader gets minimum of 45, enough to

repay loan


116

The value of a European put with dividends cannot be lower than

the Present Value of Strike price minus the Stock price plus the

Present Value of dividend

This can be seen with an example:

Table II.5.3. Upper bound of European put (with dividends)


Strike 45

PV of strike 43

PV of dividends 3

Cannot be lower than PV of strike - Stock+PV

of Dividend

43-40+3 6

If put is 5

Buy put spend 5

Borrow PV of strike get 43

Buy stock

spend

40

Take loan for PV of div. get 3

Net cash inflow now 1

Repay div. loan on getting div.

Repay other loan, from exercising put and selling stock

5.5 Upper bounds of put prices-American Options

An American put without dividends cannot be greater than or

equal to the value of the Strike price

The distinction from the European put is that the exercise can be at any time

in an American option. Therefore the put cannot have a value greater than

the Strike price.

This is illustrated below:


117

Table II.5.4. Upper bound of American put (no dividends)

(Amount in Rs) Stock 40

Strike 45

Price cannot be>

Strike

If price is 46

sell P get 46

worst case Put is exercised

your loss 45

profit 1

An American put with dividends cannot be greater than or equal

to the value of the Strike price

Three is no difference arising from the existence of dividends. The upper

limit remains as the Strike Price.

The factor to be borne in mind here is that the maximum loss to be suffered

by a put writer will be only the strike price and if he is able to invest the

present value of that straightaway he is safe. The fact that there may be

dividends on the stock is not relevant because, the dealer is not looking at the

stock at all, but only at the strike price and the price of the put

5.6 Lower bounds of put prices-American Options

An American put without dividends cannot have a price lower

than the difference between the Strike price and the Stock price

The difference from the European put may be noted. Since exercise is

instantaneous in an American option, we have to take the Strike price itself

and not its present value.

This is illustrated here under:


118

Table II.5.5 Lower bound of American put (no dividends)

(Amount in Rs) Stock 42

Strike 45

Must be greater than or equal

to

Max of 0 or Strike - Stock

it cannot be lower than 3 (45-

42)

if it is say 2.5

buy put Spend 2.5

buy stock spend42

straight gain 0.5

at

maturity

if Stock exceeds

Strike discard put

if stock is less than Strike, exercise put

trader gets minimum of 45, enough to

repay loan

The value of an American put with dividends cannot be lower

than the Present Value of Strike price minus the Stock price plus

the Present Value of dividend

This is broadly the same as the European put with dividends. The timing

and quantum of dividends might influence the exercise decision. Early

exercise is governed by the principle of time value of money and cannot be

generalized.


119

5.7 Summary The bounds of put prices also work on the no-arbitrage argument. Separate

bounds can be determined for the European and American puts, with or

without dividend.

It must be borne in mind that dividends can be considered in price

determination only if the amounts can be known in advance. Further, the

matter is complicated in American Options by the possibility of early

exercise.

The early exercise itself is based on the quantum of dividends and the timing

thereof and the intrinsic value at that time

It can be seen that the intrinsic value and the time value both play a role in

the determination of bounds.

Option bounds are useful in determining the worst outcome from certain

positions. Unless there are severe market imperfections, the bounds will

have to hold..

5.8 Key words

• Lower bound

• Upper bound

• No-arbitrage condition

• Dividends

5.9 Questions for Self- study 1) How does the American put differ from a European put in the

determination of the Upper limit of puts?


120

2) How does the possibility of early exercise affect the American

put’s lower bound when compared to the lower bound of the

European put?

3) What is the principle behind the assertion that an American put

will be at least as valuable as the European put?


121

Module 3

ADVANCED TOPICS ON OPTIONS


122

1 Option combinations


• To see some basic strategies using combinations of Options

• To understand the payoff and profit profiles from such combinations

• To understand the circumstances in which these will be used by

investors and dealers.

1.2 Introduction The advent of Options has resulted in a number of possibilities for mimicking

and creating synthetic portfolios. The pay off profile of Options in

combination creates situations very much useful for certain specific trading

requirements. Speculators use these combinations for short-term gains

Each of the combinations listed below have various profiles of profitability

and risk. They come at a cost in the form of premium, if these involve some

long positions. Written positions give a premium income in the beginning,

but involve risks.

Combinations used with positions in bonds or stock result in good synthetics

and can be used for a variety of purposes in risk management.

1.3 Straddle A straddle involves buying a call and a put at the same Exercise Price and for

the same tenure. A buyer of a straddle buys both the call and the put.

This position is illustrated below:


123

Table III.1.1. Straddle (Amount in Rs.) Strike 100

Call

premium 5

Put premium 4

initial inv. 9

Buy call buy put

End stock Call gain

Put gain

Net gain

95 0 5 -4

96 0 4 -5

97 0 3 -6

98 0 2 -7

99 0 1 -8

100 0 0 -9

101 1 0 -8

102 2 0 -7

103 3 0 -6

104 4 0 -5

105 5 0 -4

85 0 15 6

115 15 0 6

A short straddle involves selling both the call and the put. Of course the long

or short straddle can be established without buying or selling from the same

counterparty. In the above example, the call could have been bought from

one dealer and the put from another dealer.

A long straddle gains only when there is volatility and the price goes beyond

the Exercise Price in either direction beyond the total premia incurred. A

short straddle works whenever the prices remain within the band.

Another illustration is given below:


124

Table III.1.2. Straddle illustration (Amount in Rs.) Amounts

in Rs.

Strike Price 60

Call premium 1.75

Put premium 5.5

Net Cost

(-7.25)

Buy Call, Buy Put

50 0 10 2.75

51 0 9 1.75

52 0 8 0.75

53 0 7 -0.25

54 0 6 -1.25

55 0 5 -2.25

56 0 4 -3.25

57 0 3 -4.25

58 0 2 -5.25

59 0 1 -6.25

60 0 0 -7.25

61 1 0 -6.25

62 2 0 -5.25

63 3 0 -4.25

64 4 0 -3.25

65 5 0 -2.25

66 6 0 -1.25

67 7 0 -0.25

68 8 0 0.75

69 9 0 1.75

68 8 0 0.75

69 9 0 1.75

70 10 0 2.75


125

1.4 Strangle This is identical to the straddle except that the call has an Exercise Price

above the stock price and the Put an exercise price below the stock price.

It is illustrated below. As in the straddle, an initial investment is required to

go long a strangle.

A seller of a strangle banks on steady profits of high probability, but

sometimes results in risk. A long straddle is preferable to a long strangle only

if there is a premium advantage

Long Straddle

-10

-5

0

5

10

15

50

55

60

65

70

Stock Price

Gain

Cgain Pgain Net gain


126

Table III.1.3. Strangle ( Amount in Rs.) Stock 82

Call strike 85

Put strike 80

Call

premium 3

Put premium 4

initial inv. 7

Buy call buy put

End stock Call gain

Put gain

Net gain

75 0 5 -2

76 0 4 -3

77 0 3 -4

78 0 2 -5

79 0 1 -6

80 0 0 -7

81 0 0 -7

82 0 0 -7

83 0 0 -7

84 0 0 -7

85 0 0 -7

65 0 15 8

100 15 0 8

A short strangle is demonstrated below. Amounts in Rs.. Call strike 60

Call premium -1.75

Put strike 55

Put premium -2.625

Net

Inflow 4.375

Sell Call, Sell Put


127

Table III.1.4 Short strangle

Stock Call

payoff Put

payoff Net gain

50 0 -5 -0.625

51 0 -4 0.375

52 0 -3 1.375

53 0 -2 2.375

54 0 -1 3.375

55 0 0 4.375

56 0 0 4.375

57 0 0 4.375

58 0 0 4.375

59 0 0 4.375

60 0 0 4.375

61 -1 0 3.375

62 -2 0 2.375

63 -3 0 1.375

64 -4 0 0.375

65 -5 0 -0.625

Short Strangle

-6

-4

-2

0

2

4

6

50

55

60

65

70

Stock Price

Gain Cgai

n Pgain Net gain


128

1.5 Bull spreads This involves buying two calls on the same stock with the same expiry, but

with different Exercise Prices. A buyer of a bull spread buys a call with

Exercise Price below the Stock price and sells a call with Exercise Price above

the stock. The strategy has limited risk and limited profit potential. This is

illustrated below.

Table III.1.5 Bull Spreads with calls (Amount in Rs.): Stock 100

Call strike 1 95

Call strike 2 105

Call

premium 1 7

Call

premium 2 3

initial inv. 4

Buy call1 and sell call 2

End stock Call 1 gain

Call 2 gain

Tot. gain

97 2 0 -2

98 3 0 -1

99 4 0 0

100 5 0 1

101 6 0 2

102 7 0 3

103 8 0 4

104 9 0 5

105 10 0 6

106 11 -1 6

107 12 -2 6

80 0 0 -4

115 20 -10 6


129

A bull spread can also be initiated with puts. This involves writing a put at a

higher strike price and buying a put at a lower strike price. This then will

involve an initial cash inflow. This is demonstrated below:

Table III.1.6. Bull Spread with puts (Amount in Rs.) Stock 100

Put strike 1 95

Put strike 2 105

Put premium

1 3

Put premium

2 7

initial inflow 4

Buy put 1 and sell put 2

End stock Put 1 gain

Put 2 gain

Tot. gain

95 0 -10 -6

96 0 -9 -5

97 0 -8 -4

98 0 -7 -3

99 0 -6 -2

100 0 -5 -1

101 0 -4 0

102 0 -3 1

103 0 -2 2

104 0 -1 3

105 0 0 4

80 15 -25 -6

115 0 0 4

Another example is shown below:


130

Table III.1.7. Bull Spread with puts – illustration

(Amount in Rs.) Put 1 strike 50

Put 1 premium 2

Put 2 strike 55

Put 2 premium 4.375

Net Benefit 2.375

Buy Put 1, Sell Put 2

Stock Price P1Gain P2Gain Net Gain

50 0 -5 -2.625

51 0 -4 -1.625

52 0 -3 -0.625

53 0 -2 0.375

54 0 -1 1.375

55 0 0 2.375

56 0 0 2.375

57 0 0 2.375

58 0 0 2.375

59 0 0 2.375

60 0 0 2.375

Bull Spread with Puts

-6

-4

-2

0

2

4

50

52

54

56

58

60

Stock Prices

Gain P1gai

n P2gain Net Gain


131

1.6 Bear spread A bear spread with calls will involve selling a call with a lower Exercise Price

and buying a call at a higher Exercise Price. There will be no initial

investment since the call that is sold will fetch higher than the bought call.

Both profits and losses are limited. An example is shown below:

Table III.1.8. Bear Spread with calls


Call strike 1 95

Call strike 2 105

Call premium 1 7

Call premium 2 3

Initial inflow. 4

Sell call 1 and buy call 2


Call 2 gain

Tot. gain

95 0 0 4

96 -1 0 3

97 -2 0 2

98 -3 0 1

99 -4 0 0

100 -5 0 -1

101 -6 0 -2

102 -7 0 -3

103 -8 0 -4

104 -9 0 -5

105 -10 0 -6

106 -11 1 -6

80 0 0 4

115 -20 10 -6


132


Table III.1.8. Bear Spread illustration

(Amount in Rs.)

Call 1 strike 55 Call 1 premium 2.875

Call 2 strike 60 Call 2 premium 1.75

Net Benefit 1.125

Sell Call 1, Buy Call 2

Stock Call 1 gain Call 2 gain Net Gain

50 0 0 1.125

51 0 0 1.125

52 0 0 1.125

53 0 0 1.125

54 0 0 1.125

55 0 0 1.125

56 -1 0 0.125

57 -2 0 -0.875

58 -3 0 -1.875

59 -4 0 -2.875

60 -5 0 -3.875

61 -6 1 -3.875

62 -7 2 -3.875

63 -8 3 -3.875

64 -9 4 -3.875

65 -10 5 -3.875


133

A bear spread can also be carried out with puts. This will involve selling a

put with a lower Exercise Price and buying a put with a higher Exercise

Price. An initial cash outflow will be required. This is demonstrated below:

Bear spread withcalls

-12

-10

-8

-6

-4

-2

0

2

4

6

50

55

60

65

Stock Prices

Gain

C1gain C2gain Net Gain


134

Table III.1.8. Bear Spread with puts


Put strike 1 95

Put strike 2 105

Put premium 1 3

Put premium 2 7

initial inv. -4

Sell put 1 and buy put 2

End stock Put 1 gain

Put 2 gain

Tot. gain

95 0 10 6

96 0 9 5

97 0 8 4

98 0 7 3

99 0 6 2

100 0 5 1

101 0 4 0

102 0 3 -1

103 0 2 -2

104 0 1 -3

105 0 0 -4

80 -15 25 6

115 0 0 -4

1.7 Butterfly spread This involves the following:

Buying a call at a low Exercise Price

Buying a call at a higher Exercise Price

Selling two calls at an intermediate price

This strategy hopes that the price will remain within a steady range.

An example is shown below:


135

Table III.1.9 Butterfly spread


Call strike 1 95

Call strike 2 100

Call strike 3 105

Call premium 1 7

Call premium 2 4

Call premium 3 3

initial inv. -2

Buy call1 sell 2 no.s of call 2 and buy call 3


Call 2 gain

Call 3 gain

Tot. gain

95 0 0 0 -2

96 1 0 0 -1

97 2 0 0 0

98 3 0 0 1

99 4 0 0 2

100 5 0 0 3

101 6 -2 0 2

102 7 -4 0 1

103 8 -6 0 0

104 9 -8 0 -1

105 10 -10 0 -2

106 11 -12 1 -2

80 0 0 0 -2

115 20 -30 10 -2


136

-3

-2

-1

0

1

2

3

4

95 96 97 98 99 100

101

102

103

104

105

106

Tot. gain

1.8 Box spread A box spread is a combination of bull spread with calls and bear spread with

puts, at the same Exercise Price. The box spread will always pay the

difference between the high and low Exercise Price. However, the initial

investment that is needed should not be greater than the final payoff. In a

perfect market, where the prices are correctly fixed based on consistent

assumptions on volatility and risk free rates, there will be no scope to enter

into a box spread at all. But when the market has differently interpreted

assumptions for volatility and risk free interest, the prices can offer scope for

a box spread.

An example is shown below:


137

Table III.1.10. Box Spread

( Amount in Rs.) Stock 100

Call strike 1 95

Call strike 2 105

Call premium 1 7

Call premium 2 3

Put strike 1 95

Put strike 2 105

Put premium 1 3

Put premium 2 7

Buy call1 and sell call 2

Sell put 1 and buy put 2

inv. -8


Call 2 gain

Put 1 gain

Put 2 gain

Net gain

95 0 0 0 10 2

96 1 0 0 9 2

97 2 0 0 8 2

98 3 0 0 7 2

99 4 0 0 6 2

100 5 0 0 5 2

101 6 0 0 4 2

102 7 0 0 3 2

103 8 0 0 2 2

104 9 0 0 1 2

105 10 0 0 0 2

80 0 0 -15 25 2

115 20 -10 0 0 2


138


Table III.1.11. Box Spread illustration

(Amount in Rs.) Call 1 strike 50 Call 1 premium 10.25

Call 2 strike 55 Call 2 premium 8

Put 1 strike 50 Put 1 premium 3.25

Put 2 strike 55 Put 2 premium 5.5

Net Cost -4.5

Buy Call 1, Sell Call2, Sell Put 1, Buy Put 2

Stock Call 1 gain Call 2 gain Put 1 gain Put 2 gain Net gain

45 0 0 -5 10 0.5

46 0 0 -4 9 0.5

47 0 0 -3 8 0.5

48 0 0 -2 7 0.5

49 0 0 -1 6 0.5

50 0 0 0 5 0.5

51 1 0 0 4 0.5

52 2 0 0 3 0.5

53 3 0 0 2 0.5

54 4 0 0 1 0.5

55 5 0 0 0 0.5

56 6 -1 0 0 0.5

57 7 -2 0 0 0.5

58 8 -3 0 0 0.5

59 9 -4 0 0 0.5

60 10 -5 0 0 0.5


139

1.9 Summary Option combinations are important synthetic instruments which can be used

very effectively by the securities dealer. When these combinations involve

some short positions and are entered into without a covered position, they

may sometimes result in huge losses. It is therefore, necessary to frequently

monitor positions and cover and unwind them as necessary. Combinations

offer possibilities of steady profits if employed judiciously.

1.10 Key words

• Strangle

• Straddle

• Butterfly spread

• Box Spread

• Bull Spread

• Bear Spread

1.11 Questions for Self- study 1) How is a strangle different from a straddle?

Box Spread

-6

-4

-2

0

2

4

6

8

10

12

45

50

55

60

Stock Price

Gains C1gai

n C2gain P1gain P2gain Net gain


140

2) How is a bull spread will calls different from bull spread with

puts?

3) What is the logic behind the steady profit possibility under Box

spreads?

4) When will the trader employ the Butterfly spread strategy?


141

2 Principles of Option Pricing – Put call parity.


• To understand the principles behind Option pricing

• To look at the concept of intrinsic value

• To study the Binomial model and its use in Options pricing

2.2 Introduction From our understanding of the behavior of Options prices and their bounds

gained from the illustrations in the previous Chapters, we can note the

following:

1. The lower the Exercise Price, the more valuable the call

2. The difference in call prices of two calls identical except for Exercise

Price cannot exceed the difference in Exercise Price

3. Calls will be worth at least the difference between the Stock price and

the Present Value of the Exercise Price

4. The more the time till maturity, the more the call price is likely to be

5. Before expiration, a put must be equal to at least the Present Value of

the difference between the Exercise Price and the Stock Price

6. The higher the Exercise Price, the more valuable the put

7. The price difference of two puts cannot exceed the difference in the

Present Value of the Exercise Price.

8. If the stock price is 0, the value of an American call must be 0

9. The minimum value of an American call is given by either 0, or the

difference between the Stock price and Exercise Price, whichever is

greater.

10. An American call can never be worth less than the corresponding

European call.


142

11. Two American calls on the same stock having the same Exercise Price

have to be priced such that the one with the longer maturity is worth

as much or more than the one with the shorter maturity..

12. If the stock price is 0, the value of the American put must be its

Exercise Price

13. An American put is worth at least as much as its European equivalent

2.3 Some truisms about Options pricing with small illustrations The following is not supposed to be an exhaustive list of all possible

relationships in Options and pricing, but just an illustrative list to reinforce

the underlying principles.

The Lower the Exercise Price the more valuable the call Always, the call with the lower Exercise Price must have greater value. Let

us take two calls

Time Exercise Price Call price

Call A 6 months Rs.200 Rs.25

Call B 6 months Rs.195 Rs.20

In such a situation, it will be possible for a trader to sell Call A and buy Call

B for sure gains. Thus the prices have to so adjust such that Call B which

has a lower Exercise Price has a greater price than Call A.

The difference in call prices cannot exceed the difference in Exercise Prices


Call P 6 months Rs.295 Rs.20

Call Q 6 months Rs.300 Rs.14


143

Here it will be possible for the dealer to sell Call P and buy Call Q for a risk-

free profit. The prices have to adjust in such a way that the difference in the

two call prices does not exceed Rs.5, being the difference between the two

Exercise Prices

The more the time to expiration the greater the call price


Call E 2 months Rs.200 Rs.16

Call F 5 months Rs.200 Rs.15

Here any dealer can sell Call E and buy Call F, to make risk-free profits. The

prices will have to adjust to a situation where Call F, which has the greater

tenure has the greater price.

Longer the tenure the more valuable the put


Put A 3 months Rs.100 Rs.12

Put B 4 months Rs.100 Rs.11

Here any dealer can sell Put A and buy Put B, to make risk-free profits. The

prices will have to adjust to a situation where Put B, which has the greater

tenure has the greater price

Higher the Exercise Price, the more valuable the put


Put C 3 months Rs.200 Rs.11

Put D 3 months Rs.195 Rs.12

Here any dealer can sell Put D and buy Put C, to make risk-free profits. The

prices will have to adjust to a situation where Put C, which has the greater

Exercise Price has the greater price


144

The price difference between two American puts cannot exceed the difference in Exercise Prices


American Put E 3 months Rs.200 Rs.4

American Put F 3 months Rs.205 Rs.10

Here any dealer can sell Put F and buy Put E, to make risk-free profits. The

prices will have to adjust to a situation where the two prices do not exceed

the difference between the Exercise Prices. The difference in the two put

prices must not exceed the 5 (205-200).

The price difference between two European puts cannot exceed the difference in the Present Values of their Exercise Prices

Time PV of Exercise Price of Rs.100 Call price

European Put G 3 months Rs.95.12 Rs.4

European Put H 3 months Rs.98.88 Rs.10

Here any dealer can sell Put H and buy Put G, to make risk-free profits. The

prices will have to adjust to a situation where the two prices do not exceed

the difference between the present value of the Exercise Prices. The

difference in put prices must not exceed 3.76 ( 98.88-95.12).

2.4 Put call parity For a given call price, the corresponding put price for the same Exercise Price

and same tenure can be found out. This called the Put-Call parity rule.

Let us take the example of two portfolios. The first portfolio has a long

position in stock and a long put. The second portfolio has a long call and an

investment in risk-free bonds to the extent of the present value of the strike

price. This investment in the present value of the strike price will grow to


145

become the strike price at the end of the tenure. At expiry the stock can end

up greater than the strike price or equal to or below it. The performance of

the two portfolios in these eventualities is mapped below:

Table III.2.1 Put call parity

First Portfolio

Second Portfolio

Stock +Put Call + PV of strike

At expiry

If Stock is less than Ex.Price

If Stock is greater than Ex.Price

Port. A

Stock Stock price at end

Stock price at end

Put

Ex.Price-

Stock price 0

TOTAL Exercise Price

Stock Price at end

Port B

Calls 0

Stock Price at end – Ex.Price

Bonds Ex.Price Ex.Price

TOTAL Exercise Price

Stock Price at end

So original values should be

equal

Stock

+Put = Call +PV of strike

Thus we establish that the value of a Put will be


146

Put = Call + PV of strike price – Stock price

Call =Stock price + Put price – PV of strike price

In respect of American calls, the Put-call parity rule is more difficult to

determine. The following example shows the steps. Let us take two

portfolios. Portfolio P consists of 1 European call and Cash worth the Strike

Price. Portfolio Q consists of 1 American put and 1 share

The pay off profile of the two portfolios is shown below:

EVENTUALITY Payoff

IF STOCK AT

END >STRIKE

PRICE

(STOCK AT

END-STRIKE

PRICE) +

STRIKE

PRICEert

PORTFOLIO P

I European Call + Cash

worth the Strike Price

IF STOCK AT

END<STRIKE

PRICE

STRIKE

PRICEert

IF STOCK AT

END>STRIKE

PRICE

0 +STOCK AT

END

PORTFOLIO Q

I American Put + I

share

IF STOCK AT

END<STRIKE

PRICE

(STRIKE PRICE-

STOCK AT END)

+ STOCK AT

END

So Portfolio P is MAX (1) Stock at end –Strike + Strike* ert, or

(2) Strike Price*ert

Portfolio Q is MAX (1) Stock Price at

end or (2) Strike Price


147

Portfolio P is more valuable than Portfolio Q. In case of stock price being

higher than exercise price, Portfolio P is greater by the interest earned on the

Strike Price. If the stock price is lower than exercise price, the value is

higher than Portfolio Q again because of the interest earned on the Strike

Price.

If PORTFOLIO Q is exercised early, its value then will be X, whereas at that

point of time PORTFOLIO P will be worth Strike Price plus interest earned

up to that point (Strike Price compounded for the period up to the time of

exercise). Again PORTFOLIO P will be more valuable than PORTFOLIO Q.

So European Call + Strike Price > American Put + Stock

European call –American Put > Stock – Strike (Equation 1)

We know from the Put-Call parity rule for European Options that

European Call + Present Value of Strike =European Put +Stock

Or

European Call – European Put = Stock – Present Value of Strike (Equation 2)

American call is at least as valuable as the European call

Combining Equations 1 and 2

(Stock – Exercise Price) < (American Call – American Put) < (Stock – PV of Strike)

This gives us the range of prices for American Puts given the corresponding

Call prices and vice-versa

2.5 Exercise of the American Call early The advantage that an American option has over the European option is that

the former can be exercised at any time during the currency of the contract.


148

However, in the case of calls it can be shown that it is not optional for an

American call holder to exercise the option before the expiry of the contract.

There are only two alternatives if one were to exercise the call option before

expiry:

1. One can keep the shares obtained out of the exercise as an investment.

2. One can sell the shares in the market straightway.

The first choice is inferior to retaining the option itself. If one exercises the

option and keeps the shares as an investment, one will have to incur capital

expenditure for buying the shares. If, instead, the option is retained, one can

exercise the option at any time one likes and there is no need for upfront

investment. Besides, by exercising and holding, one is taking a risk of the

investment value falling, whereas if the option is retained, it acts as an

insurance against such a fall, such that one need not exercise the call at all.

This shows that whatever the eventual price, exercising the option for

investment purposes is not optimal.

The second choice relates to selling the shares in the market immediately

after the exercise of the option. This implies that the option is sufficiently in

the money and for fear of a fall in the market the option could be exercised

straightway and the shares obtained out of the exercise immediately sold off.

But if this is the intention, the call itself could be sold off in the market for

the same pay-off. An American in-the-money call has to have the minimum

value of (stock price minus Exercise Price), so the advantage of exercising,

taking the shares and then selling them off can also be obtained by just

selling the call. By the second choice one gets (Stock – Exercise Price), which

will always be the minimum price of an American option. So early exercise is

of no use in this case as well.


149

Thus, whatever the circumstances, early exercise of American call Options is

not optional.

The cost that you paid entering into for the option is irrelevant because it is a

sunk cost and is the same for both the alternatives.

Because of the relatively low volumes in the derivative market it may

sometimes not be possible for one to dispose of a call option at will. But on a

conceptual front, one is looking at perfect market conditions.

Illustration

Let us take the case of an American call option purchased for Rs.5 with an

Exercise Price of Rs.50 and time to expiry of 3 months. Let us assume that

the option becomes in the money (i.e. the stock price exceeds the strike price)

and the stock price is Rs.60. The alternatives open to the buyer of the option

are:

1. Exercise and sell the stock. He will pay Rs.50 per share for the

exercise and get Rs.60 for the sale thereby making a profit of Rs.10.

2. Sell the call itself. The call will now be priced at a minimum of Rs.10

(stock price – Exercise Price). This is because if the option were priced

lower say at Rs.8, all anyone has to do is to buy the option at 8 and

exercise straightway to get Rs.10 (60-50) and make a risk free profit of

Rs.2. So the in-the-money American option price has to be necessarily

Rs.10. It can of course be greater than Rs.10 taking into account the

volatility of the underlying asset.


150

So coming back to our example, the minimum price that we will get by selling

the call itself is Rs.10. Whereas the maximum price one will get by selling

the share is Rs10. So, early exercise does not make sense.

2.6 Exercise of the American put early The position is different when it comes to exercise of the American put.

When the holder of the put finds that it is sufficiently in the money, he may

as well exercise it early. Here, he will get the proceeds of the exercise (the

sale value) immediately and will therefore enjoy the time value of money

before the maturity. This is in contrast to the position of the American call

holder, who had to pay out the money and therefore lose the time value

before maturity. As far the holder of the American put is concerned, the only

consideration is as to whether the stock has gone down enough. Further, he

has to make up his mind as to whether the volatility of the stock is coming

down. If the volatility is still there, it might be worthwhile to hold on to the

put option, unless, of course it has already become sufficiently in the money.

For example suppose X holds a put option on a scrip with Exercise Price of

Rs.50. Suppose a month before the maturity, the scrip is trading at 40. X

has to decide

1. Whether the Rs.10 that he will make by exercising the put now will be

enough as per his portfolio profit strategy.

2. What is the extent of volatility now in the scrip? If the volatility is high

even now, it may be worthwhile holding on in the expectation of a

further fall.

The above two issues have to be balanced and a decision arrived at. The

actual exercise is a matter of judgment. The point to be noted, however, is

that, early exercise is not a non-optional solution, as in the case of calls.


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2.7 Summary The pricing of Options follows certain principles. These principles are based

upon no-arbitrage conditions. If disparities exist in prices alert dealers will

buy and sell in the market in such a way that they reap risk-free profits. As

more and more dealers exploit the disequilibrium, the prices adjust to their

correct levels.

Several principles have been laid down in this unit based upon these no

arbitrage conditions. The intricate relationships between the expiry time in

the asset price and option values result in a number of valid principles.

For a given call price there must be a corresponding put price on the same

underlying asset with the same Exercise Price. This can again be established

using the no arbitrage condition. This principle is called put-call parity.

It can be shown by using principles of time value of money that it is not

optimal to exercise the American call earlier than maturity. Although,

American calls carry this right it will be not worthwhile for the holder to

exercise this right before maturity. However, in case of necessity the dealer

can sell the American Option in the market .

This principle does not hold for an American put because of the favorable

time value of money. Here it is optimal to exercise the American put before

maturity if it is sufficiently in the money.

2.8 Key words

• Put-Call Parity

• Arbitrage

• Optimal exercise


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2.9 Questions for Self- study 1. If the difference between the Exercise Prices of two American puts is

Rs.10, can the two puts have prices of Rs.26 and Rs.12 ? Explain.

2. Why is it not optimal to exercise the American call before maturity?

3. Does the put call parity rule apply to American Options?


153

3 The Binomial model for pricing of Options


• To understand the principles governing the Binomial model of Options

pricing

• To understand the differences in approach between no-arbitrage

pricing and risk-neutral valuation

• To understand the concept of implied volatility or chance under the

risk-neutral valuation

• To take the Binomial model forward to a two-period framework

• To generally understand how the model works if the number of periods

is increased

3.2 Introduction We have seen how the option prices work in relation to the time to expiry and

Exercise Price. The impact of the stock price on the pricing has also been

seen. We have set upper and lower bounds of option prices. Now we have to

attempt to pinpoint a price which should be appropriate for an option given

the time to maturity, strike price and stock price.

For the purpose the Binomial model has been found to be an intuitive

explanation of the pricing of Options. While the assumptions of the Binomial

Model might appear to be far-fetched., it can be established that these do not

mar the ability of the model to come to almost correct prices. Studies have

also shown that if the Binomial Model is carried to a large number of periods

it will correspond to the Black –Scholes model of Option pricing.


154

3.3 Binomial one-period model We can look at the Binomial one-period model with an example.

Let us assume the present stock price to be Rs.106. Let us also assume that

over a one year horizon the price of the stock will become either Rs.120 or

Rs.90. Options are available in the market with an Exercise Price of Rs.100.

The above facts indicate the following:

Range of Stock prices Range of call payoff

Up Rs.120 Rs.20

Down Rs.90 0

From the above let us work out for a combination of shares and calls in such

a way that the total value remains the same both on the Up and Down

movements. This involves a ratio of shares long and calls short. A

combination of long two shares and short three calls will result in the

following final payoff.

Stock Calls Total

Up Rs.240 - Rs.60 Rs.180

Down Rs.180 0 Rs.180

So regardless of what happens in the market (whether the prices go up or

down) the total payoff from the combination is always Rs.180. The

combination of two shares long and three calls short was arrived at using

simple arithmetic. Formally this proportion is called the Optimal Hedge

Ratio and is found out by the following formula:

CU – CD

-----------

SU – SD

Where CU stands for the payoff of the call for the up movement

CD stands for the payoff of the call for the down movement


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SU stands for the payoff of the stock for the up movement

SD stands for the payoff of the stock for the down movement

In the above example the formula gives us an answer of 2/3 which means 3

calls for 2 shares.

So the final value is always Rs.180 after 1 year. It may be recalled that as a

principal assumption to the Binomial Model there can be only two scenarios

and either case the portfolio is Rs.180. A portfolio that always yields Rs.180

is risk free in nature and its present value is its discounted figure based upon

a risk free rate of return. Rs.180 discounted for one year at 8% risk free rate

of interest (assumed) gives a figure of Rs.166.67.

The portfolios value today ought to be Rs.166.67. The portfolio consists of two

shares long and two calls short. The two shares long cost Rs.212 (the present

stock price of Rs.106 multiplied by 2). The difference between Rs.212 and the

portfolio value of Rs.166.67 gives us the value of the three calls shorted. This

value is Rs.45.33 and therefore the value of 1 call is Rs.15.11.

The same answer could have been obtained by following a different approach

called The Risk Neutral Valuation. The theory of Risk Neutrality says that

investors are indifferent to the actual probabilities of payoff and are only

concerned with getting a payoff equal to the risk free rate of return. The two

movements of the asset values are not assigned any probabilities but the

implied probability for their movements can be determined by using the risk

free rate of return.

The implied probability can be determined by using the following equation in

respect of the stocks.

120p + 90 (1 – p) = 106 * e0.08 * 1


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Where p refers to the implied chance of up movement

( 1 – p ) refers to the implied chance for the down movement .

e is the natural logarithm.

The formula can be put as follows:

The above formula seeks to equate the implied chances of up and down

movements to the risk free rate of return on the original value of the stock

calculated using continuous compounding. The equation yields a p value of

0.816. Therefore (1 – p) is Rs.0.184. Since we are interested in the call prices

and since we have already determined that the call payoff will be either Rs.20

or 0 the following equation can be used to determine the value of the call.

( 20 * p + 0 *1 –p)

= 20 * 0.816 + 0 = 16.32.

This figure of 16.32 is the value of the call after one year. When this is

discounted at the risk free return of 8% we get Rs.15.11.

It can thus be seen that both the optimal hedge approach and the risk

neutral approach yield the same answer.

3.4 Binomial two-period model In the above illustration we had assumed that we have only one intermittent

period where prices could change. When we extend this theory to more than

one period, there will obviously be more branches to the tree. However, as

per the assumption of the model, the quantum of increase and decrease

remains a fixed percentage of the start point. Thus, if the original one-period

assumption had two possibilities – increase by 10% or decrease by 5%, the

p= (1 + risk free return)- (1- down rate) ------------------------------------------- (1+ up rate) – (1+ down rate)


157

same percentage increase and decrease will apply to all the nodes in the

other periods as well.

Let us work on a 2-period model with an example:

Let us assume that the stock price is Rs.100 and that it can either go up by

10% or come down by 5% in a single period. For simplicity, we can take one

period to be 1 year. We are trying to value a call option on the stock with an

Exercise Price of Rs.105. The call is assumed to be valid for 2 years. The

risk-free rate of return is 8%.

The first step is to draw the payoff diagram for the stock

The calculations are straightforward based on the percentages of up and

down movements indicated.

100

110

95

104.5

121

90.25


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The corresponding payoff position in respect of calls will be as follows. The

payoff has been calculated by reducing the strike price from the final price. If

the final price is less than the strike price the payoff is 0.

The procedure is to first determine the Options payoff in the end and work

backwards node by node. The three branches at end of year 2 have payoff of

Rs.16, 0 and 0. The top most branch will be in-the-money and fetch Rs.16.

The other two branches end up out-of-the-money and hence have a value of 0.

(Please recall that Options cannot have a negative value, since they can be

discarded when not in profits).

?

?

?

0

16

0

p= (1 + risk free return)- (1- down rate) ------------------------------------------- (1+ up rate) – (1+ down rate)


159

The call value at the top branch of Year 1 can be calculated using the payoff

estimations for year 2. But before that the implied probabilities under the

risk-neutral valuation should be computed. This is found by using the

formula

The formula yields a value of p of 0.87, and (1-p) to be 0.13.

Using this we can calculate the payoff in the top branch of Year 1 as (16

*0.87) + (0*.13) =13.92, discounted for 1 year at the risk-free rate of 8%. This

gives a figure of 12.89 as the figure for the top portion of Year 1.

We can calculate the value of the bottom node in the same way. In this

example, the two possible payoff figures at end of Year 2 are both 0, and

hence the value of the bottom node at end of Year 1 will also have to be 0. In

case the two payoff figures in Year 2 are positive, we can calculate their

corresponding value for the node in Year 1 by attributing “p” values and (1-p)

values and then discounting by the risk-free rate for 1 year.

The figure looks as follows now:

?

13.92

0


160

The p value is 0.87 and (1-p) is 0.13. Applying these to the Year 1 payoff

figures and discounting for 1 year yields Rs.11.21as follows:

(13.92*0.87) + (0*0.13) = 12.11. This discounted for 1 year at 8% gives 11.21.

So the value of the call at inception is Rs.11.21.

3.5 Extension of the principle to greater number of periods This principle can be used for calculating the value for any number of

periods. We begin with the last period, estimate the payoff for the option

then, find the p value and (1-p) value and then discount the possible payoff

by one period. This process is continued for every possible node in the

binomial lattice. Working backwards, we ultimately come to the value at the

start.

If the period of possible price change is shorter than 1 year, we have to adjust

the discount rate accordingly. Thus, if the changes are recorded for every

half-year, the applicable discount rate for bringing back the possible payoff

figures is 4%, being half the annual rate.

Lastly, while it may look far-fetched that we can have only two possible

movements of prices under the model, it should be noted the we can achieve

more accuracy by just reducing the duration of each period, thereby

increasing the number of nodes.

Beyond a point it is difficult to calculate the above by hand and we may

require the assistance of a software program. There are a number of

spreadsheet models designed for calculating the Binomial Options price.

percentages.


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The major difficulty confronting the analyst will be to assert that estimated

levels of up and down movements will remain the same for all periods to

come.

3.6 Summary An understanding of pricing of Options is best had with the Binomial model.

The Binomial model assumes that stock prices can have only two possible

movements in a period, by a specified percentage up or specified percentage

down.

Valuation under the Binomial model can be done using the no-arbitrage

argument or the risk-neutral approach. Under the former, a portfolio of long

stocks and short calls is created, in such a way that the final payoff is

identical for both the possibilities. The combination is arrived at by

calculating an Optimal Hedge Ratio. When the final payoff is identical and

does not involve any risk, its value initially will be its discounted value at the

risk-free rate. This initial value consists of the value of both stock and calls.

The value of stock being known is deducted to get the value of the calls.

The second approach to valuation involves taking a risk-neutral assumption.

Under this, it is taken that the investor is concerned only about getting the

risk-free return, and a payoff that has an estimated value of the risk-free

return, will have certain implied chances. The value of these implied chances

are calculated by taking the final payoff to be the original amount

compounded by the risk-free return, and then finding the value of the two

branch payoff figures to suit this final payoff. Using these the payoff of the

calls is calculated and discounted back to the beginning. Both the approaches

give identical answers.


162

An understanding of the Binomial model is essential to enable us to

appreciate the working of the option pricing mechanism. There are certain

limitations to the model but nevertheless it has been found to be very useful

as a valuation tool. The challenge before the analyst, however, would be to

attribute specific percentage chances of up and down movements, applicable

for all periods to come.

3.7 Key words

• Binomial Model

• Risk Neutral

• No Arbitrage

• Implied chance

• Risk free return

3.8 Questions for Self- study 1. What is the principle behind risk neutral valuation?

2. What are the components of a typical portfolio used for determining

the no – arbitrage binomial price?

3. Are the assumptions of the binomial model far fetched?

4. What is the discount rate used for bringing the binomial payoff back to

start date?


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4 The Black-Scholes model

4.1 Objectives The objectives of this unit are :

• To understand the principles underlying the Black Scholes model.

• To understand the principles behind the log normal distribution

• To understand the Black Scholes formula .

• To get a first hand account of calculation of option prices using the

model.

4.2 Introduction The Black Scholes model is a Nobel-prize winning attempt to define option

prices. As we have seen the Binomial approach gives us fair indication of

option prices subject to certain conditions. A different approach has been

used in formulating the Black Scholes model.

The model uses ideas from other sciences in determining the value of the

Option. First, share prices have been known to follow the principle of

random walk and stock returns have been found to be best represented by a

log-normal distribution. The model takes into account several important

inputs the most important of which is the stock volatility. Intuitively, a

greater level of volatility will result in a greater option price. Similarly, the

longer the time available the greater the option value.

The Black-Scholes model, is essentially useful only for determining European

calls and puts. The model directly arrives at the European call price, from

which it will be possible calculate the European put price using principles of

put-call parity.


164

The formula discussed later in this unit is difficult to derive, but it is also

quite intuitive in application. The advent of special calculators and

spreadsheet –based packages has resulted in universal use of this model by

traders and investors.

The model works on certain assumptions which some critics say are

unrealistic. However, as we know most of the models in finance are based on

some extreme assumptions which need to be fine-tuned by the user

depending upon his specific circumstances. If the model is otherwise capable

of reflecting the correct situation its assumptions should not be held against

it.

Empirical evidence of the Black Scholes model is divided. Ultimately the

model will work only on our inputs being right. If the correct price is not

reflected by the model it could be because the wrong input has been fed in.

The model has five specific inputs – stock price, strike price, risk free rate of

return, tenure and volatility. Each of these inputs has different impacts on

the option price as demonstrated later.

4.3 Some preliminary ideas The Binomial model is a discrete- time model, with specific time intervals.

The Black-Scholes model, however, takes into account an infinite number of

sub-intervals, on a continuous time basis. It can be established that if taken

for sufficiently long periods with multiple interim periods, the conclusions

drawn by the Binomial model tally with those arrived at by the Black-Scholes

model.

The Black-Scholes model has been developed for European Options and can

easily be applied to American calls on non-dividend paying stocks.


165

The derivation of the model is complex and involves advanced mathematics.

No attempt has been made to show the derivation here. However, the logic

behind certain aspects of the model merits attention.

Lognormal distribution

A stock is supposed to follow a Brownian model and the distribution of the

stock returns are hence lognormal. Under this, the least value of a stock

return is -100%. This is superior to the idea under a pure normal

distribution that the price can be less than -100%. Lognormal distributions

can be seen to be skewed to the right and do not follow the bell shape of a

normal distribution.

Return, variance and price

The value of an option as brought out by the Black-Scholes model is

independent of the expected return. The model does take into account the

stock prices, but the expected rate of return itself is not considered. Stock

prices are assumed to follow a “random walk” and their estimations are based

on volatility estimates. Secondly, the model also assumes that the variance is

proportional to the time. The estimate of variance is not expected to change

during the tenure of the contract.

4.4 Assumptions under the model The Black-Scholes formula has certain inherent assumptions:

1. There are no dividends on the stock

2. There are no transaction costs

3. The short-term risk-free interest is known and is constant during the

lifetime of the Option

4. Short selling of stock is permitted

5. Call option can be exercised only on expiration

6. Trading takes place continuously and stock prices move randomly


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7. Stock prices follow the rules of a Brownian process and stock returns

are best explained by the lognormal distribution.

4.5 The formula European Call price (C) is found by

N(d1), N(d2) = cumulative normal probability

σ = annualized standard deviation of the continuously

compounded return on the stock

cr = continuously compounded risk-free rate

S = Stock price now

E = Exercise Price

In=logarithm to the base è

T = time to maturity

4.6 Illustration Let us take the stock price to be Rs..20 and the strike price to be also Rs, 20.

The tenure of the option is 3 months. The risk-free rate is 12% and the

variance of returns is 0.16. The value of the Option under the Black-Scholes

method can be calculated as under:

Here, d1 = ln (20/20) + (0.12 + 0.16/2) *0.25

------------------------------------------

(0.40) (0.50)

Solving we get this to be 0.25

Then d2 = d1 – 0.4 √0.25

TddT

/2)T(rln(S/E)d

where)N(dEe)SN(dC

12

2c

1

2Tr

1c

σ

σ

σ

−=

++=

−= −


167

We get this to be 0.05

N(d1) and N(d2) represent areas under the normal distribution. From the

normal distribution tables, we get

N(d1) for a 0.25 area to be 0.5987

For N(d2) for a 0.05 area we get 0.5199

The value of the call then is

(20 *0.5987) – ( 20 è(-0.12*0.25) * 0.5199)

Solving we get the value to be Rs.1.88.

Another example is given below.

Here the stock price is Rs.30 and the Exercise Price is Rs.25. The time to go

is 3 months (0.25). The risk-free return is 5% and the standard deviation is

0.45. ( the variance is 0.2025)

Here, d1= ln (30/25) + (0.05 + 0.2025/2)*0.25

--------------------------------------------

0.45*0.25

We get the answer to be 0.978.

Then d2 = d1 – 0.45 √0.25

We get this to be 0.753

Nd1 for 0.978 = 0.836

Nd2 for 0.753 = 0.773

With this the value is

(30 *0.836) – ( 25 è(-0.05*0.25) * 0.773) = 6

The value of the call is Rs.6


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4.7 The model inputs The Black-Scholes works on five inputs, all crucial to the determination of

Option values;

• The Stock Price

• The Strike Price

• The risk-free return

• Volatility

• Time to expire

4.8 The Black-Scholes calculator Both Traders and practitioners use the Black-Scholes calculators for

determining the theoretical prices of Options. It should be recalled that the

basic formula calculates the value of a European call. Using the principles of

put-call parity, it is possible to compute the corresponding put prices for the

same expiry and strike price.

There are several versions of the calculator available. One version is

available for download from the internet and is applicable for direct use with

Microsoft Excel.

Numa Financial Systems Ltd

email: [email protected]

web: http://www.numa.com/

Taking our two illustrations above, we can compute the option prices using

the calculator as follows:

Input data

Stock price Rs.20

Strike price Rs.20

Time to go 3 month (0.25)

Risk-free rate 12%

Volatility (standard deviation) 0.4


169

Inputting this data, we get Rs.1.88 as the value of the call, exactly the same

result as we got by the long calculation.

4.9 Impact of variables on Options pricing The impact of various inputs on the final Options price as per the Black-

Scholes model is given below:

Stock price

The value of the call increases as the Stock price increases. This is

demonstrated below, for the illustration already covered. We had found that

the Options price was Rs.1.88 for a stock price of Rs.20. The Options price for

various other stock prices are below. Please note that the other variables are

the same:

Stock price

Options Price

18.0000 0.8894

19.0000 1.3340

21.0000 2.5275

22.0000 3.2568

23.0000 4.0556

Exercise Price

The value of the call decreases with increase in Exercise Price. This is

demonstrated below using the same inputs.

Strike Price

Call Values

18.0000 3.0860

19.0000 2.4358

21.0000 1.4258

22.0000 1.0584

23.0000 0.7719


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Tenure

When the tenure increases the value of the option goes up. This is intuitive

in that the more the time available the more the value of the option. This is

shown below. In the original case the time was 0.25 ( 3 months). Let us find

the values for other time periods.

Time Option Value

0.1500 1.4105

0.2000 1.6582

0.3000 2.0902

0.3500 2.2849

0.4000 2.4695

Risk-free rate of interest

As the risk-free rate goes up, the Option value goes up. The principle is that

increase in risk-free rate results in reducing the Present Value of the

Exercise Price, thereby increasing the Option price. This is shown below.

Please recall that the original option value of Rs.1.88 was obtained with a

risk-free rate of 0.12 ( 12%).

Interest Option value

0.1000 1.8326

0.1100 1.8575

0.1300 1.9078

0.1400 1.9333

0.1500 1.9589


171

Volatility

As the volatility goes up, the value of the Option goes up. An option derives

its value from uncertainty and the greater the level of uncertainty the greater

the value of the option. This goes back to the basic feature of Options as

instruments. An Option has an edge over the asset itself in that it takes only

the favorable swings and can discard the unfavorable swings. As such the

option holder will hope for greater chances of a swing. In the original

illustration we had taken the standard deviation to be 0.4. The impact of

changes to the standard deviation on the option prices are given below:

Std. Deviation

Option price

0.3000 1.4968

0.3500 1.6894

0.4500 2.0761

0.5000 2.2698

0.5500 2.4634

4.10 Summary The Black Scholes model is considered as a very elegant piece of research

into option prices. The model uses ideas from the Brownian motion and other

theories based on “random walk” . The model involves certain inherent

assumptions and yields the European call price. Among the assumptions the

most notable are the one relating to the log normal distribution of the stock

price. Studies have shown that this assumption is quite valid. Another

assumption regarding the constant volatility during the tenure of the option,

is open to question. The formula for calculating the Black Scholes model

price is relatively simple. The presence of special calculators and

spreadsheet solutions make the task very easy for traders and investors.


172

4.11 Key words

• Brownian Motion

• Log normal distribution

• Random walk

• Volatility

• Area under normal curve

4.12 Questions for Self- study 1) What are the various assumptions under the Black-Scholes

model?

2) Give two important inputs in the Black-Scholes model and their

impact on Option value?

3) Does the Black-Scholes model give the same result as the

Binomial model for Option prices?

5 Volatility and Implied Volatility from the Black-

Scholes model

5.1 Objectives The objectives of this module are :

• To understand the concept of volatility

• To understand the implication of volatility in option prices

• To study the concept of implied volatility

5.2 Introduction Among the several inputs into the Black Scholes model volatility is the most

crucial. This refers to the extent to which the stock price can change during

the tenure of the contract. An option carries greater inherent value with

greater volatility.


173

Estimation of volatility is a controversial subject. If volatility is based upon

historical data, the implicit assumption is that the future will behave in the

same way as the past. Even if that is taken as a valid assumption the

question remains as to how many months in the past do we have to go to for

ascertaining the past volatility. The figures of volatility could be vastly

different if taken for say a three-year period compared to say a six-month

period. The implication of this is that whatever we feed in as volatility will

result in a corresponding price right, or wrong.

Market analysts use a slightly different technique for estimating volatility.

Four of the inputs in the Black-Scholes model are known. By feeding in these

and also the price prevalent in the market, the market’s interpretation of

volatility can be derived.

5.3 Importance of Volatility and the concept of Implied volatility Four of the five inputs into the Black-Scholes model are straightforward.

They are the stock price, the strike price, the risk-free rate of interest and the

tenure. These parameters are readily available and can be fed in directly.

There could be some question marks about the correct interest rate to be

taken. But in the context of the overall decision, this is a relatively minor

choice.

The fifth input – Volatility is difficult to estimate. The model wants the

likely swing that the stock price can take over the period of time under

consideration. For this the standard approach is to take a historical

volatility. Here, we take the standard deviation of returns over a period of

time and expect this to be the volatility. However the period of reckoning can

make a difference in the estimate. The figure of volatility could be totally

different say if a 1-year historical volatility is taken as compared to a


174

situation if a 3-month historical volatility is taken. Consequently, there could

be substantial differences in the Option values as calculated.

Market observers seek to solve this problem by studying the extent of

volatility that the market seems to be imputing to the prices. This can be

arrived at by studying the four clear inputs with the Option prices prevalent

in the market. A trial and error estimation gives us an idea as to the

volatility that is implied in the market actions and is called Implied

Volatility. Many calculators have features which calculate this directly given

the four standard inputs and the Options price.

5.4 A discussion The impact of volatility can be assessed from various angles. From the angle

of checking the validity of the model itself, the test would involve comparing

the historical volatility of a scrip or a basket, with the implied volatility

arising out of the going option price. The latter is used taking the Black-

Scholes assumptions to be valid for the case in point. Alternatively, the

option price that should have been there given a historical volatility can be

compared with the actual option price for testing the significance of the

differences, if any.

It is possible to estimate the underlying asset volatility on the basis of past

experience. In many ways this issue is similar to the problem in the

estimation of returns and comparison under the Capital Assets Pricing

Model. In analysing stock prices, it can be noticed that there are two types of

volatility that they suffer from. The first is the result of the inherent risk of

its projects not going through to expectations, the extent of its own growth

potential, the competition from within and outside the country and changes

in its management and financing patterns. These are Company-specific risks

that stem occasionally from industry-specific risks and are called by the


175

technical name “unsystematic risk”. It is demonstrated by portfolio theory

that unsystematic risk could be diversified away by combining it with other

stock that have a different element of such risk, in a manner that will result

in the portfolio combination having a combined unsystematic risk that is

negligible.

The other type of risk is the “systematic” risk or market risk. Here,

depending on the overall market movements, the scrip will move up or down

depending on its “connection” with the market. We are all aware that there

are certain incidents that affect the entire stock market such as changes in

budget policy, fall in agricultural production, changes in RBI policy and

fluctuations in foreign exchange position. Apart from these, there is

something called “market sentiment” which takes the overall stock prices up

or down from time to time. Depending upon each stock’s relationship with

the overall market, it will be affected by some degree. Some stocks move

exactly with the market with some moving more than proportionately on the

same side of the market, while some others moving inversely.

From the angle of the trader, it is necessary to go by some historical volatility

as a starting point. True, some embellishments will be needed to such

historical data, but to have a basis, it will necessarily have to begin with the

past. The question often arises as to how far back one should go to get an

estimate of volatility that is good enough for the future. The issues involved

are discussed below:

• The first question is whether there is sufficient evidence to show that

the past volatility will hold in the future. If the market is perfect and

information asymmetry is minimal, one would expect all market

players to have homogenous expectations. In such a scenario, price

movements have to follow the historical volatility pattern, unless there


176

are changes in circumstances. In the semi-strong form Efficient

Market Hypothesis, all the reported information is also within the

reach of the market and the price is reflective of that.

• The next question is the period to be considered for the purpose. Here

there cannot be unanimity. Just as in the case of estimation of Beta,

the period to be reckoned becomes a matter of subjectivity. Unless the

market is highly volatile in the short run, it may make more sense to

take the long-term (say 3 years) volatility as the basis. However,

because changes in Government policy have a great bearing on the

systematic risk, it may be sometimes safer to take a 1-year horizon

• Further, the question that needs to be addressed and understood is as

to the applicability of these estimations in practice. This brings to us

the question as to whether the market follows these estimates of

volatility or is arbitrary in its behavior. The implication of such a

suggestion is that the Black-Scholes Model itself has then no

applicability to the price determination model.

In the initial stages of Options being introduced, the general perception is

that options prices will not correspond to a model-determined price. This is

because the writers charge a special premium and there are few

combinations going around which act as an effective mechanism for checking

radical price changes. However, as the market becomes a little experienced

(as in India now), it becomes increasingly adept at using option combinations

and other synthetic instruments, which make sure that the prices are based

on uniform assumptions. Since the Black-Scholes model describes Options-

price determination in the most scientific way, we will have to take the

postulates of the model to be right unless evidence is overwhelmingly

different. As such, the estimates of volatility by the Options market has to be


177

based on volatility estimate for the spot asset market, with some

modifications for the period of the option.

The actual data of stock prices, call prices and put prices of Hindustan Lever

Ltd, for a short period in 2006 has been shown below, to show that the put-

call parity rule has not held for most dates. Further, the question as to

whether the prices indicated have the correct volatility depends on the

volatility estimates we have in mind. The implied volatility during this

period has not been consistent.


178

Table III.4.1. Pricing and put-call parity of HLL

(Amount in Rs.) HLL

Date Stock price

Call price for strike 215

put price for strike 215

Call-Put

Stock price- Ex.price

2-Aug-06 228.15 33.15 16.4 16.75 13.15

3-Aug-06 224.7 30.25 17 13.25 9.7

4-Aug-06 220.5 27.05 18.05 9 5.5

7-Aug-06 219.45 25.3 17.5 7.8 4.45

8-Aug-06 223.05 27.5 16.15 11.35 8.05

9-Aug-06 224.05 27.35 15.05 12.3 9.05

10-Aug-06 223.45 26.25 14.55 11.7 8.45

11-Aug-06 223.2 25.35 13.95 11.4 8.2

14-Aug-06 227.15 27.2 12 15.2 12.15

16-Aug-06 230.5 28.85 10.35 18.5 15.5

17-Aug-06 234.9 31.65 8.8 22.85 19.9

18-Aug-06 237.95 33.45 7.6 25.85 22.95

21-Aug-06 236.35 31.45 7.3 24.15 21.35

22-Aug-06 232.6 28.3 7.95 20.35 17.6

23-Aug-06 229.85 25.9 8.35 17.55 14.85

24-Aug-06 231.8 26.8 7.35 19.45 16.8

25-Aug-06 236.6 30.2 6 24.2 21.6

28-Aug-06 240.3 32.55 4.75 27.8 25.3

29-Aug-06 234.9 28.3 5.95 22.35 19.9

30-Aug-06 236.5 29.05 5.15 23.9 21.5

31-Aug-06 234.45 27.1 5.25 21.85 19.45

5.5 Summary Empirical evidence shows that actual call values of stock-based Options in

the sample period are significantly different from the theoretical prices based

on volatility for various periods indicated. The degree of differences has

varied from study to study depending on the markets studied and the nature

of price movements. There appears to be good correlation between the


179

theoretical prices and the actual prices. But this only indicates the direction

of the movement of change and not the magnitude. Intuitively the direction

can be more easily predicted than the magnitude.

Many studies on the subject of implied volatility have shown that actual call

values of the Index are quite different from the indications of volatility.

Further it appears that the market relies more on the volatility estimate of a

1-year historical period in arriving at the call values.

5.6 Key words

• Volatility

• Implied volatility

• Historical volatility

5.7 Questions for Self- study 1) From a given data of option prices how can you calculate the implied

volatility?

2) Will the historical volatility match the implied volatility at all?

3) Is it true to say that the implied volatility will ultimately become the

historical volatility?

4) Does the difference between the historical volatility and implied

volatility show that the market is arbitrary?


180

Module 4

OTHER DERIVATIVES AND RISK MANAGMENT


181

1 Introduction to Options Greeks and Basic Delta

Hedging


• To understand the concept of Option Greeks.

• To understand the implication of various Greek measures.

• To understand the concept of Delta Hedging.

1.2 Introduction We have seen that the Black Scholes model is an elegant exposition of the

determination of option prices. Five specific inputs are required for getting

the option price out of the model. These are the stock price, the strike price,

the tenure, risk free rate of return and volatile return. The impact of changes

in call prices on account of changes in these inputs are determined by Option

Greeks.

Market analysts and traders find Option Greeks very useful in formulating

specific strategies. The impact of specific inputs on the price is a very useful

piece of information in drawing up a plan for a portfolio of Options. While the

derivation of these Greeks is mathematical, it can be readily taken from an

Options calculator.

1.3 Delta and uses The extent to which Option price changes in response to a change in stock

prices is measured by Delta. It is the change in the Option premium

expected out of a small change in the stock price.

Delta is measured as follows


182

The Delta of a call approaches 0 as it becomes more and more out-of-the-

money. A call that is at the money will hover around the 0.5 level. An in-the-

money call approaches 1.

The Delta of a put approaches 0 as it becomes out-of-the-money

An in-the-money put approaches (-1)

The purchase of a call which has a Delta of 0.5 is equivalent to buying half

stock by borrowing at the risk-free rate of interest

The purchase of a put with Delta (-0.2) is equivalent to selling short 0.20

stock and investing in risk-free assets

The Delta of a put is equal to the Delta of a call (-1)

Delta mimics the number of shares required to match the change in prices of

Options. Suppose a call has a Delta of 0.80 it works like having 0.80 stock

and a change in stock price of Re.1 results in change in call value of Rs.0.8

In European Options, the absolute value of a call and put total 1.

SC

c ∂∂

=∆SP

p ∂∂

=∆

Delta of a call Delta of a put


183

In terms of the Black-Scholes model, the call Delta is equal to N(d1).

Delta can be directly calculated from Option calculators.

1.4 Delta hedging The following equations can be used in respect of Delta

• Call = Buying Delta times stock by borrowing

• Put = Selling Delta times stock and investing in bonds

From the call formula we can rearrange the relationship as under:

Short call + Buying Delta times stock by borrowing = 0

This relationship is used by traders for Delta-Hedging their short positions in

calls. They continuously buy Delta times the stock (with borrowing) and

together the position will neutralize each other. However, Delta holds only

for small changes in value. Further, Delta position needs to be rebalanced

constantly. An illustration is shown below.

We take the case of a stock which is going at Rs.42. A dealer has written a

call on the stock with an Exercise Price of Rs.40. He has earned a premium

of Rs.3.53 on this written call. He wants to embark upon a Delta hedge to

cover this short position. The risk-free rate of interest is 10%.

Let us further assume that the trader wants to Delta Hedge by only

rebalancing once in a week. There are 13 weeks to go to the end of the

contract.

It may be noted in this context that a Delta hedge will work better if

rebalanced very frequently. A weekly rebalancing may serve the purpose, by

and large, but cannot be expected to be fool proof.


184

The full chart of prices of the stock at the end of each of the 13 weeks to

expiration, along with corresponding call price and call delta is shown below.

Please note that the time has been specified in decimals with 13 weeks

corresponding to 0.25, 12 weeks corresponding to 0.23 and so on.

Table IV.1.1. Delta data

(Amount in Rs.) Weeks to go

Underlying price Strike Volatility Interest

rate Time to expiry

Theoretical value Delta

13 42 40 0.2 0.1 0.25 3.5295228 0.7846199

12 41.5 40 0.2 0.1 0.2307692 3.04044517 0.7490096

11 42.5 40 0.2 0.1 0.2115385 3.72173953 0.8251148

10 41.75 40 0.2 0.1 0.1923077 3.01254878 0.7737713

9 42.25 40 0.2 0.1 0.1730769 3.2988242 0.8178858

8 43 40 0.2 0.1 0.1538462 3.82440488 0.876425

7 42.5 40 0.2 0.1 0.1346154 3.28069483 0.8523

6 43.25 40 0.2 0.1 0.1153846 3.83147374 0.9120885

5 43.75 40 0.2 0.1 0.0961538 4.1926855 0.9485565

4 44 40 0.2 0.1 0.0769231 4.33506007 0.9702544

3 43.5 40 0.2 0.1 0.0576923 3.75412925 0.9706311

2 42.75 40 0.2 0.1 0.0384615 2.92699687 0.9650667

1 43 40 0.2 0.1 0.0192308 3.07814875 0.9964273

0 43 40 0.2 0.1 0.001 3.0039998 1

The dealer buys shares to the extent of Delta times the position, at the end of

each week. The Delta keeps changing week after week in the light of the

changing stock price, and the reducing period. He buys this Delta times stock

by borrowing at the interest rate of 10%.

The position is shown below:


185

Table V.1.2. Delta Hedging (Amount in Rs.)

Week Stock Price Delta

Needed Position In

Shares

No. Bought Cost Cum.Cost Interest

0 42 0.78462 78461.99 78461.99 3295404 3295404 6337.3

1 41.5 0.74901 74900.96 -3561.03 -147783 3153958 6065.3

2 42.5 0.825115 82511.48 7610.52 323447 3483471 6699.0

3 41.75 0.773771 77377.13 -5134.35 -214359 3275811 6299.6

4 42.25 0.817886 81788.58 4411.45 186384 3468494 6670.2

5 43 0.876425 87642.5 5853.92 251719 3726883 7167.1

6 42.5 0.8523 85230 -2412.50 -102531 3631518 6983.7

7 43.25 0.912088 91208.85 5978.85 258585 3897087 7494.4

8 43.75 0.948556 94855.65 3646.80 159547 4064129 7815.6

9 44 0.970254 97025.44 2169.79 95471 4167415 8014.3

10 43.5 0.970631 97063.11 37.68 1639 4177069 8032.8

11 42.75 0.965067 96506.67 -556.44 -23788 4161313 8002.5

12 43 0.996427 99642.73 3136.06 134851 4304167 8277.2

13 43 1 100000 357.27 15363 4327807

The changes in prices result in change in the number of shares held for Delta

hedging. Occasionally, the dealer sells shares from his holding if the new

Delta at any point of time entails only a lesser holding. Interest is calculated

for every week and shown in the last column.

A summary of his final position is given below.

The call having ended in the money, his Delta has become quite close to 1 at

the end of the 12th week. Let us assume that the price remains at this level

at the end of week 13 as well. Now the dealer is able to use his holding of

100000 shares to cover his short position. He will get the call strike price of


186

Rs.40 per share. The future value of the premium he received up front

Rs.353000 (100000*3.53), works out to Rs.357097.

Against this, the dealer has incurred a cumulative cost of Rs.4327807 for

buying the shares as indicated in various weeks shown above. His net

position then is as under:

gain loss

Call Strike received

4000000

4327807 Amount paid

for shares

Value of

premium 357097

4357097 4327807

This shows that the loss has been lower than the gains and that that he has

gained a little in the Delta Hedge. Had the time interval of rebalancing been

smaller the dealer would have been able to rebalance every day instead of

every week, and the position would have been more equal. Since the

rebalancing cannot be continuous there will always be a tracking error.


187

1.5 Gamma, Theta, Vega and Rho

Gamma

Gamma is the second derivative of the option premium in relation to the

stock price. Thus, Gamma is the first derivative of Delta to the stock price.

Gamma tells us the extent to which the Option portfolio needs to be adjusted

to conform to a Delta hedged position. A gamma of 0 indicates that the Delta

is not very sensitive to price changes. If Gamma is small, delta changes only

slowly. The Gamma of a call is equal to the Gamma of a corresponding put at

the same exercise price. Gamma is never negative. Adjustments to keep the

portfolio delta-neutral need to be done only infrequently in such a case.

Traders sometimes follow a strategy called delta-gamma hedging. This

results in being delta hedge and also being simultaneously gamma hedged.

This strategy involves setting up of a further call position. When the

portfolio is delta-gamma hedged, it need not be adjusted frequently.

The relationship can be shown as under:

SSPG

SSCG

pp

cc

∂∆∂

=∂∂

=

∂∆∂

=∂∂

=

2

2

2

2


188

Theta Theta is the sensitivity of the call price to the time remaining to maturity.

Sine the time available can only get shorter, Theta is represented as a

negative number. Any passage of time is not beneficial to the buyer of an

option, but is beneficial to the seller.

The Theta for calls and puts can be calculated as follows:

A comparative position of the Delta, Gamma and Theta is shown below:

Position Delta Theta Gamma

Long call + - +

Long Put - - +

Short call - + -

Short put + + -

Vega Vega is the derivative of the Option price with reference to volatility. An

Option derives value from volatility. A Vega of 0.2 indicates that a change of

volatility by 1% will result in 0.2% change in option prices. Vega is measured

as follows:

)(22

)(22

2

)(5.

2

)(5.

21

21

dNrKet

eS

dNrKet

eS

rtd

p

rtd

c

−−

−−

+=Θ

−−=Θ

πσ

πσ


189

Rho Rho is the derivative of the Option price with reference to the risk-free rate of

interest, used as an input in the model. It is the least important of the

Greeks. Unless an option has a very long tenure, a small interest rate

change is unlike to have any great impact on the option price.

1.5 Summary Option Greeks constitute an important source of strategic information for the

dealer. These measures – Delta, Theta, Rho, Vega and Gamma – show the

impact of specific inputs on the option price. The likely change in the option

price as a result of a small change in these inputs is measured .

The most actively used Greek is Delta. Many traders use Delta for hedging

their short call exposures. Sometimes asset managers who have a portfolio of

Options decide to Delta-hedge to avoid losses from a portion of their portfolio.

Further alert investors can sometimes use the Delta technique for exploiting

wrong implied volatility in the market.. However, the assumption here is

that wrong implied volatility will soon get corrected to the levels anticipated

by the trader.

π2vega

)(5.0 21detS −

=


190

1.6 Key words

• Delta

• Theta

• Rho

• Gamma

• Vega

• Delta Hedging

• Delta Hedging Arbitrage

1.7 Questions for Self- study 1) How does Delta hedging safeguard a short call position?

2) Will Delta hedging work for all changes in value?

3) Is Vega important? How can it be used by traders?

4) How is Gamma different from Delta? What is the principle behind

Delta Gamma hedge?


191

2 Interest Rate Derivatives and Eurodollar Derivatives


• To understand the function of interest rate Derivatives.

• To appreciate the functioning of Forward Rate Agreements.

• To understand the distinction between caps and flows.

• To understand the use of Eurodollar Futures.

2.2 Introduction We have seen the operation and broad use of general Derivatives. We now

look at the use of these instruments in managing a portfolio of fixed income

securities. These assets carry considerable risk which can be covered with the

use of special Derivatives. Interest rates are a matter of great uncertainty

and these Derivatives seek to cover this risk.

Before we go into categories of interest rate Derivatives it is necessary to

understand the broad relationship between bond prices and yield. The Fixed

Income securities market uses the concept of yield-to-maturity to determine

return. The yield-to-maturity (YTM) is the internal rate of return of the bond

cash flow. The initial price paid for the bond, the periodic coupons and the

end price on redemption constitute the bond flows. The market is never able

to predict the movement of YTMs across various maturities and across

different credit ratings. Hence holders of bonds are exposed to considerable

interest rate risk.

The bond price has an inverse relationship with YTM. When the market

YTM goes up the bond prices come down and vice versa. This stems out of

the operation of the internal rate of return.


192

Interest rate Derivatives seek to cover some of the exposure that bond

managers have. First it is necessary for the portfolio manager to determine

the specific direction of movement of interest rates that she fears and then

choose appropriate interest rate Derivatives.

In India, Interest rate Derivatives are allowed to be traded in the National

Stock Exchange. Rules regarding trading and settlement can be had from

their website www.nse-india.com

2.3 T Bill and T Bond Futures These instruments are very popular in the United States. T Bills are

generally short term and T Bonds will be of greater duration. However, the

principles governing their prices are the same.

The U.S T Bill sells at a discount from the par value and is represented by

the discount yield. Discount yield refers to the following formula:

Thus a 90- day Bill quoted at say 98.50 will have a yield of 6 %. The day

count convention for these bills generally has 360 days in the year.

The discount yield cannot be directly compared with other investments,

because it relates the income to the par value, as against the usual practice of

relating it to the price paid.

The standard face value of T Bills is Rs.1 million.

From the discount yield the price to be paid for purchase can be determined

as follows:

(Par Value – Market Value) multiplied by 360------------------------------------------------ ---- Par Value 90


193

Suppose T Bill Futures are bought at Rs.93 ( discount yield of 7%), the price

to be paid then will be

Rs.982500, using the above formula for a 90-day period.

If interest rates rise to 7.25%, the new price will be Rs.981875, using the

same formula but now inputting 7.25% as the discount yield.

The results are intuitive in that the price of the bond comes down when the

yield goes up.

Speculators buy T-Bond Futures when they expect the interest rates to fall

and vice-versa

2.4 Hedging with T Bills and T-notes A corporate house expects to have Rs.20 million for short-term investment.

The amount will be available to it in 3-months time. The Corporate

Treasurer is apprehensive that the interest rates might fall in the meantime,

resulting in a fall in the expected interest when the investment is made. He

therefore goes in for long T-Bill/T-Note Futures. As noted already, these

notes will rise in value when the interest rates fall.

The present interest is 7%. The price then will be Rs.982500 for every Rs.1

million contract, and therefore Rs.19,650,000 ,for the Rs.20 million contract.

If as expected by the Treasurer the interest rate falls to say 6.5%, the new

price of the T Bill/T- Note will be Rs.983750 for a Rs.1 million contract. The

Discount yield *90 Price = Face Value multiplied by (1- -------------------------)

360


194

total price available to the Treasurer on the Rs.20 million contract will be

Rs.19,675,000, gaining Rs.25,000 on the Futures contract.

Now the treasurer can invest his surplus Rs.20 mn. in the market and along

with the additional Rs.25000 got out of the Futures contract, he can makeup

for the loss in interest rates.

Just as we saw in our discussion on regular Futures, the treasurer would

have lost on the Futures contract if the interest rates had gone up instead of

coming down. In such an eventuality he would have been better off without a

Futures contract at all. He could then have participated in the higher interest

by investing the Rs.20 mn. as and when available at the higher interest rate.

While he can still do so, the loss in the Futures contract drags down his

profits. However, in trying to freeze an interest yield the Futures results in

unfavorable conditions on opposite movements.

2.5 Eurodollar Derivatives The word “Eurodollars” refers to dollar amounts held outside the United

States. Investors in US can deposit their surplus amounts lying in the US

banks to a Eurodollar Bank at a marginally higher rate of interest. The Euro

bank, in turn does not get access to the amount directly, but can lend it

others needing Dollar amounts at a rate of interest lower than that prevalent

in the US for such loans. The borrower will be able to use Dollar funds now

lying in the credit of the Euro bank, from the US banks.

The London Inter-bank Offer Rate ( LIBOR) is the arithmetic average of the

rates at which six major Institutions in London would be willing to deposit or

lend their dollar funds.


195

Eurodollar Futures are popular in the US. The yield on a Eurodollar Futures

contract is calculated on a 360-day basis, and is calculated as a discount from

the par value.

Eurodollar Futures are exchange settled based on the differences in prices.

As per market practice, the seller of the Eurodollar Futures is the Fixed-

interest payer. He agrees to pay a specified % on the Notional Principal in

exchange for getting back a floating rate, to be arrived at after 3 months.

The buyer of the Eurodollar Futures is the fixed interest receiver and agrees

to pay the floating rate prevalent after 3 months in exchange for the fixed

interest on a Notional principal. Thus if the price today is 97 (corresponding

to an annual interest of 3%), the seller of Eurodollar Futures banks on the

interest rate going up. If the interest rates go up to say 4%, the price will be

96 and he will gain. Interest rates going up amount to floating rates going

up. Thus the seller is trading a fixed interest for receiving a floating interest.

Conversely, the buyer of the Futures hopes that the interest rates will come

down. If it falls to say 2%, the price will be come 98 and he will gain. So the

buyer agrees to give the new interest rate in exchange for getting a fixed

interest.

Like other Futures contracts, Eurodollar Futures contracts are subject to

mark-to-market

2.6 Forward Rate Agreements For a short-term fear of interest rate changes, A Forward Rate Agreement

(FRA) can offer a good solution. A seller of an FRA will receive a fixed rate of

interest on a Notional Principal over the specified period in exchange for

giving a floating rate of interest.. A buyer of an FRA will receive a floating

rate of interest against paying a fixed interest.


196

The concept of Notional Principal is very important for FRAs. The amount is

not exchanged between the counterparties but forms the basis for

determining the payoff.

Let us take the case of a Company which has been sanctioned a loan of Rs.10

million. The loan will become operative in 3 months and the Company is

obliged to pay a floating rate of interest on the loan at 1% over the SBI PLR

as on the last date of this quarter. Suppose the loan is to be taken with effect

from 01/04/07, the SBI PLR as at 31/03/07 will be reckoned and a 1% addition

made thereto to. This will be the applicable interest rate on the loan for the 3

months from 01/04/07 to 30/06/07. The interest for the next 3-month period

(01/07/07 to 30/09/07) will be reckoned by taking the SBI PLR rate as on

30/06/07 and adding 1% to it.

At this juncture, it should be noted that the determination of bench mark

interest rate and the percentage addition/deduction thereto are matters of

contractual expediency and will vary from case to case. Also a specific

contract can stipulate as to how the bench mark floating rate is to be

determined for purposes of interest calculation. In the above example we

have assumed that SBIPLR is reckoned as of the last date of a quarter.

Instead, some contracts could stipulate that the rate ought to be the average

of the three months. It is a purely a contractual stipulation and is not rigid.

Coming back to our example our Company fears that interest rate in the

markets are likely to go up in the short run. Particularly, it fears that

interest for the quarter 01/04/07 to 30/06/07,will shoot up but expects interest

rates to stabilize in the subsequent quarters to lower levels. Therefore it

wants to safeguard against payment of high interest in the ensuing quarter.


197

For this purpose it can enter into an FRA promising to pay a fixed rate of

interest against receiving the benchmark floating rate (plus or minus a few

basis points).

In this example let us assume that the current SBIPLR (as on 01/01/07) is

8%. So if interest rates were to remain at this level the Company will have to

pay 9 %( SBIPLR plus 1%) on the loan for the period starting 01/04/07.

However, since it fears a rise in interest rates, the actual interest payable

could be much higher. An FRA with another counter party will ensure that

the Company pays 9% fixed on the notional principal, against receiving the

SBIPLR plus 1% from the counter party.

Before we map the payoff under the FRA in various scenarios, it is necessary

to appreciate that the FRA terms can be customized in any way. For instance

the notional principal need not be the Rs.10 million that the Company has in

mind but could be a smaller figure. Further the exchange payment in

floating rate need not exactly match the Company’s potential liability under

the loan. Thus the FRA could provide for a floating rate receipt of say

SBIPLR plus 0.5%.

The FRA payoff under different rates of ultimate interest is given below.

Table V.2.1. FRA payoff (Amount in Rs.) Interest rate %

Rate +1%

Interest due FRA in FRA out Net

payment 6 7 175000 175000 225000 225000

7 8 200000 200000 225000 225000

8 9 225000 225000 225000 225000

9 10 250000 250000 225000 225000

10 11 275000 275000 225000 225000

11 12 300000 300000 225000 225000


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The first column shows the interest rates in the end. The corresponding rate

+1% is the next column. Before entering into the FRA, the Company had

contracted to pay the applicable floating rate +1% to its bank. This is the

amount shown in Column 3. In terms of the FRA the Company will receive

the SBIPLR +1% from the counterparty, in exchange of paying 9% fixed.

These amounts are indicated in the next two columns. The amounts have

been calculated using the 3 month period as the basis. The last column

indicates the net amount incurred by the Company from the deal, which is

the total of the interest it pays to its bank plus the amount it pays on the

FRA minus the amount it receives from the FRA.

In the above example, the FRA has succeeded in freezing the amount it has to

pay out, in every eventuality.

There is one more aspect to FRAs. The net payment shown in the last

column is due only on the expiry of the 3-month period. For instance the

interest rates as shown in Column 1 is as on 31/03/07, it is applicable for the

period from 01/04/07 to 30/06/07, and becomes actually due on 30/06/07.

Thus, if the exchange of payment has to be done immediately on

determination of date – 31/03/07 – it has be the Present Value of the amount

as shown in the last column, discounted at the ruling floating rate.

2.7 Caps A cap is an option by which the buyer gets the right to get the difference

between the actual interest rate and a strike interest rate on a notional

principal. For example let us assume A buys a European cap from B with

strike interest rate of 8% on a notional principal of Rs.1 million, for a 3 month

period. If the interest after the three month period becomes 10%, B will be

obliged to pay A Rs.20,000(2% on Rs.1 million). The principal is purely


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notional and is not exchanged. The payoff position under various interest

rates, is mapped below.

Table V.2.2. Cap payoff (Amount in Rs.) Cap struck at 8%

interest rate

CAP inflow

6% 0

7% 0

8% 0

9% 10000

10% 20000

11% 30000

12% 40000

It may be noted that a cap is similar to a call option. It does not force the

buyer to a level of interest but gives him the cushion in case the interest

rates go up. Thus this is superior to buying an FRA . Here the buyer gets

the best of both the worlds. However, like any other option a cap will also

entail payment of a premium.

2.8 Floors A floor is an option by which the buyer gets the right to get the difference

between the strike interest rate and the actual interest rate on a notional

principal. For example let us assume P buys a European floor from Q with

strike interest rate of 8% on a notional principal of Rs.1 million, for a 3 month

period. If the interest after the three month period becomes 7% Q will be

obliged to pay P Rs.10,000(1% on Rs.1 million). The principal is purely

notional and is not exchanged. The payoff position under various interest

rates, is mapped below.


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Table V.2.3. Floor payoff (Amount in Rs.) Floor struck at 8%

Interest rate

CAP inflow

6% 20000

7% 10000

8% 0

9% 0

10% 0

11% 0

12% 0

It may be noted that a floor is similar to a put option. It does not force the

buyer to a level of interest but gives him the cushion in case the interest

rates come down. Thus this is superior to selling an FRA . Here the buyer

gets the best of both the worlds. However, like any other option a floor will

also entail payment of a premium.

2.9 Collars A collar is a combination of calls and puts. Generally a collar involves buying

one option and selling the other. Thus the premium received on selling one

option goes to reasonably offset the premium paid on buying the other. In an

ideal situation the two premia will offset each other resulting in what is

called a costless collar.

The payoff of a typical collar is mapped below. For the purpose of the table

below, we assume that the buyer of the collar pays a premium on the call put

and gets a premium on the floor sold. The notional principal in both the cases

is Rs.1 million and the collar is costless.


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Table V.2.4. Collar payoff

(Amount in Rs.)

Costless collar struck at 8%

Long Call and Short Put

Interest rate

CAP flow

FLOOR flow

Total COLLAR

flow 6% 0 -20000 -20000

7% 0 -10000 -10000

8% 0 0 0

9% 10000 0 10000

10% 20000 0 20000

11% 30000 0 30000

12% 40000 0 40000

2.10 Summary Derivatives are very useful in interest rate risk management. These

basically cater to two broad category of users – holders of bond portfolios and

regular dealers fearing interest rate changes. The Bond Portfolio manager is

concerned about possible depletion in net value consequent to changes in

interest rate. Every decline in market interest rate results in the bond

portfolio value going up and every increase in value results in the value

coming down. Interest rate Derivatives can be used to cover these risks.

General borrowers or lenders of money can also use interest rate Derivatives

to cover themselves against possible increases or decline in interest rates.

T-Bill Futures constitute the most basic of Interest rate Derivatives. These

are traded in many stock exchanges the world over. Similar to this in

function is the Eurodollar Futures which are based on the London Inter-bank

Offer Rate (LIBOR). Forward Rate Agreements (FRAs) constitute another


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category of such Derivatives. FRAs are similar to Futures in terms of

functions and enable the participant to freeze levels of interest. Pure interest

rate Options like caps and floors are also useful in capturing any unforeseen

value while at the same time covering the risks.

2.11 Key words

• Bond prices

• Yield to maturity

• T-Bill Futures

• Forward Rate Agreements

• Eurodollar Futures

• Caps

• Floors

• Collars

2.12 Questions for Self- study 1) What is the broad difference between buying a call and going

short the T-Bill Futures?

2) Does the FRA always give the buyer his chosen rate of return?

3) What is the principle behind discount yields?

4) How is hedging using T-Bill Futures different from hedging

using FRAs?


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3 Swaps


• To understand the concept of swaps

• To understand the economics behind swaps

• To understand the principles behind a currency swap

• To understand the principles behind valuation of a swap

• To understand unwinding of swaps

3.2 Introduction Companies are committed to certain specific liabilities in their everyday

management. Occasionally, they seek to come out of certain risky obligations

by exchanging these with others who have a mirror-image requirement. The

reasons for this “cleaning –up” could vary from Company to Company. Many

reasons exist like seeking to reduce the overall risk exposure, wanting to shift

the exposure to a new type and to generally diversify one’s portfolio.

Most of the other Derivatives and particularly the interest rate Derivatives

we saw work only on the basis of unilateral requirements. Their success

assumes a certain level of market participation and volumes in the market.

Swaps, on the other hand require a tailor-made agreement with a specific

counterparty with a similar requirement, but in the opposite direction.

There are many types of swaps. We look at interest rate swaps and currency

swaps and then seek to understand their valuation.

Swaps can also be thought as a series of Forward Rate Agreements. This is

especially so in the case of Plain Vanilla Interest Rate Swaps.


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3.3 Plain Vanilla Interest Rate Swaps By far the most common and the most easily understood swap is the Plain

Vanilla Interest Rate Swap. We can work on the methodology of this swap

with an example with certain assumptions.

Let us take the case of a Company X which has an outstanding loan of Rs.10

million, which will get fully repaid in 4 years’ time. The loan had been taken

2 years back based on a floating rate of interest based on SBI PLR. The SBI

PLR at the time when the loan was taken was 9% and currently it is 10.5%.

The Company will be more comfortable if the interest rate were fixed at

around 10%. Of course, the Company realizes that a fixed interest rate will

not give it the benefit of any fall in the floating rate, but taking into account

its general level of profits and other inflows, the Company would like to

freeze the interest payment at 10%.

Another Company – let us call it Company Y- has a similar loan amount, but

its liability is fixed in nature. Its policy requires it to align its interest

liability to market conditions. In the process, it is prepared to take the risk of

interest rates going up as well.

If the two companies – X and Y- enter into an agreement by which they

exchange interest payments on a notional principal for a specified period, the

purposes of both are met. In the above example, Company X will make

periodic payments to Company Y payments on a fixed basis against getting

interest on the basis of the benchmark interest rate agreed on, from

Company Y. The position can be roughly drawn up as follows


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Company X

• To pay Floating interest to its lending bank

• Will get this floating interest from the counterparty Company Y

• Will pay Company Y a fixed interest rate of 10%

In the process Company X will benefit if interest rates are actually higher

than 10%. For instance if the interest rates are 12%, Company Y will pay it

12 % . Against this, a payment of 10% fixed will be made by Company X to

Company Y. Of course the two payments will be netted. However, if the

interest rates fall to say 9%, Company X will still have to pay 10% fixed to

Company Y, against receiving 9% from Company Y. Thus, its liability will

remain fixed at 10%.

Company X

• To pay Fixed interest of 10% to its lending bank

• Will get this fixed interest from the counterparty Company Y

• Will pay Company X a floating interest rate of SBIPLR.

In the process Company Y will benefit if interest rates are actually lower

than 10%. For instance if the interest rates are 9%, Company X will pay it 10

% . Against this, a payment of 9% floating will be made by Company Y to

Company X. Of course the two payments will be netted. Further, if the

interest rates rise to say 12%, Company Y will still have to pay 12% floating

to Company X, against receiving 10% from Company X. Thus, its liability

will be floating

In the process of the swap Company A has succeeded in converting its

floating rate liability to fixed rate liability, and Company Y has converted its

fixed rate liability to a floating rate liability.


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The actual terms can be varied based on the convenience of the two

counterparties. Thus, it could be that Company X may pay a fixed of 10.25%

against receiving SBI PLR -.25%, etc.

3.4 Exploiting disequilibrium in interest quotes – the Spread differential.

In the actual market, plain vanilla swaps are used for exploiting lack of

balance in interest rate quotes. This can be seen from the following

illustration:

Company P wants a loan of Rs.10 million. Its bankers have told the Company

that a fixed interest loan can be sanctioned at 10% interest, while a floating

interest rate can be sanctioned at the half-yearly SBI PLR + 1 %.

Let us assume that another Company – Company P- is also looking for a

Rs.10 million loan. Its bankers have given it a quote of 11% for a fixed

interest loan and SBI PLR + 3 % for a floating interest loan.

The position can be mapped as follows:

Type of loan Company P Company Q

Fixed 10% 11%

Floating SBIPLR + 1% SBIPLR + 3%

Here it can be observed that Company Q pays more than Company P in

respect of both Fixed and Floating loans. Obviously, Company P is better

credit-rated. However, Company Q is not paying proportionately the same

level higher both the type of loans. Thus, for the floating loan, Company Q

has to pay more excess relatively that it has to pay for the fixed rate loan. It

will have to pay 1% extra on a fixed rate loan, but will have to pay 2% extra if

it chooses a floating rate loan. This is called the Spread Differential and can

be exploited by the two counterparties if certain other conditions exist.


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To carry the example forward, let us assume that both Companies P and Q

are indifferent as to the type of loan that they take. They are comfortable as

of now with both a Fixed or a Floating loan. Since Company Q gets the fixed

rate loan cheaper relatively, it can borrow on the Fixed segment and

Company P can borrow on the other segment – in this case floating. This is

called the principle of Comparative Advantage. They then can swap their

flows. One such possible swap can be that Company P will pay Company Q

9.25% as its swap flows, in return for getting SBIPLR + 1% from Company Q.

The position will then be as follows:

Table V.3.1. Plain Vanilla Swap

Company P Company Q

Borrow

Floating loan

Borrow fixed

loan

Pay interest

to bank

-SBIPLR +

1%

Pay interest

to bank

-11%

For swap pay

to Company

Q

-9.25% For swap pay

to Company

P

-SBIPLR

+1%

For swap get

from

Company Q

SBIPLR +1% For swap get

from

Company P

9.25%

TOTAL -9.25% -SBIPLR

+2.75%

In the process Company P ends up having a fixed liability of 9.25% interest.

If it had borrowed directly from its bankers without the swap, an fixed

interest loan would have cost it 10%. So Company P has gained 0.75%


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Company Q ends up having a floating rate liability and pays interest of

SBIPLR + 2.75%. This is 0.25% lower than what it would have had to pay if

it had borrowed the floating rate directly.

In the above case, the Spread Differential of 1% has been shared by

Companies P and Q with P getting 0.75% and Q getting 0.25%. This is just a

matter of negotiation. On a different level of sharing, the swap inflows and

outflows will get rearranged to come to the new level. The two counterparties

know what they have to pay to their lending banks. Having decided the

sharing ratio, they know what they have to end up having to pay. These two

figures enable the actual swap flows to be drawn up.

It should be noted that this scheme will work only if the two counterparties

are indifferent as to the type of interest rates that they are going to end up

paying. If either of the parties is particular about a type of interest, then the

swap will work only if the Comparative Advantage position allows this.

Thus, if Company P was particular about having only a floating rate liability,

the swap would not have worked, since it entails P to end up having a fixed

liability.

The actual swap flows can be drawn up in any manner to suit the final

position. In practice, only the differences are netted. The principal is

notional and not transferred.

Interest rate swaps might also involve an intermediary for arranging the deal

and to sometimes take over the default risk. In such circumstances, the

intermediary also has to be paid and this will be reduced from the total

available Spread Differential in a suitable manner. So the amounts to be

shared by the two counterparties and the intermediary bank have to be

determined in advance to enable the mapping of the sharing and flow.


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3.5 Currency Swaps The motivation for a currency sap arises partly out of comparative advantage

and partly out of the need to manage specific flows in a country. Thus, a U.K

firm having is operations in India will like to have an Indian loan which can

be repaid over the next six years, say, with the receipts from its operations in

India. Let us assume that the US Company has estimated its receipts from

Indian operations to range from Rs.60 million to Rs.75 million every year. So

it is in a position to comfortably repay up to Rs.60 million every year.

Suppose there is a mirror-image Indian Company having some operations in

UK. This Company is looking for a UK loan which can be repaid out of the

proceeds of its operations in UK. Suppose this Company has estimated its

UK receipts over the next six years to be in the tune of 0.7 million pounds to

1 million pounds. It can comfortably repay any loan involving annual

repayments of around 0.7 million pounds.

Let us assume that the exchange rate between the two currencies as of today

is Rupees 85 to one pound.

The UK Company will find it easier to borrow in UK rather than in India.

Its credit rating in UK will be higher and regulatory procedures will not

create any difficulties if the loan is to be in pounds. For the same reason, the

Indian Company will like to have its loan in Indian rupees. However, the

actual requirement of the two companies is in the opposite currency. Hence,

one feasible solution would be for them to borrow in the currency of their

convenience (comparative advantage) and then swap it with each other. In a

currency swap the principal is also exchanged. The procedure amounts to

borrowing in local currency and depositing it, while borrowing the foreign

currency. In the end both the loan and the deposit mature and are closed


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Indian

Co. Borrow in

Rupees Rs.240 mn.

UK Co.

Borrow in

pounds

2.82 mn.

Pounds

The amounts of loan correspond to the exchange rates prevailing as of now.

Even the interest rates as determined have a bearing on the exchange rates

on account of the principle of interest rate parity.

Table V.2.2. Currency Swap Flows Yr.1 Yr.2 Yr.3 Yr.4 Yr.5 Yr.6

Indian Company ( in Rs.) million

Borrowed 240.00

given to UK Co. -240

paid to bank -24.00 -24.00 -24.00 -24.00 -24.00 -24.00

got from UK Co. 24.00 24.00 24.00 24.00 24.00 24.00

got from UK Co. 240

Paid back loan -240

UK Company (in pounds) million

Borrowed 2.82

given to Indian Co. -2.82

paid to bank -0.20 -0.20 -0.20 -0.20 -0.20 -0.20

got from IndianCo. 0.20 0.20 0.20 0.20 0.20 0.20

got from Indian Co. 2.82

Paid back loan -2.82


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The net effect is that the Indian Company has been able to get funds from

UK (2.82 million pounds) at a rate of interest of 7%. The UK firm has been

able to get a loan of Rs.240 million Indian rupees at a rate of interest of 10%.

At the end of the tenure both these loans have to be repaid.

There are many variations to the basic currency swap model shown above.

One of the interest swap payments could be floating. Or the swap could be

through an intermediary and based on swap quotations. The easiest way of

mapping a swap flows is to treat it as a deposit by the Company of loan

proceeds in its own currency and borrowing in foreign currency. At the end

both the deposit and loan are closed.

Currency swaps will be useful only if the amount to be paid as interest as a

result of it is lower than what the Company itself would have been able to

borrow in the foreign currency. The market convention is to have swap

quotes in all-in terms. All-in calculations involve finding the internal rate of

return of the stream of flows. Unless the all-in cost of the swap is less than

direct all-in borrowing cost in the foreign currency, a swap is not worthwhile

3.6 Valuing swaps and unwinding The value of a swap at the beginning is generally 0. The present value of the

floating rate segment and the fixed rate segments are expected to match. If

these were not expected to match, the counterparties would not have entered

into the deal at all. There could, of course, be differences in perception, but

the calculations by each must indicate that the value is 0.

At any point of time, the remaining floating interest payments need to be

discounted back at the relevant floating rate itself to come to the original

value. Since the interest factor and the discounting factor are the same, this

brings us back to the principal. The fixed payments need to be discounted at


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the relevant floating rate from time to time. Thus, the discounting rate for

period 1 will be different from that in period 2, because of the difference in

floating rates expected in the two periods.

For instance let us take a situation of a loan for Rs.10000, where the floating

rates are 7.2% ( applicable to interest payment to be made one year from

now). Let us assume interest payments to be annual. The floating rates are

expected to be 7.2% after 1 year( applicable to interest payment to be made 2

years from now) and then are expected to rise to 8% ( applicable to interest

payment to be made 3 years from now). Thus the interest payments on such

a floating loan are expected to be Rs.720, Rs.720 and Rs.800 for Years 1, 2

and 3. The principal is to be repaid in Yr.3. The flows are as follows:

Table V.2.3. Swap flows (Amount in Rs.) yr.1 yr.2 yr.3

Interest

payment 720 720 830

principal

repay 10000

Total 720 720 10830

Here, yr.1 flows have to be discounted back at 7.2%, year 2 flows at 7.2% and

yr.3 flows at 8%.

The principal repayment is also to be discounted back at 8%. This results in

the value coming back to the original figure of Rs.10000. Now if the fixed

interest payments of Rs.800 and the final principal of Rs.10000 are

discounted back at the relevant expected floating rate, we get a figure of

Rs.9945. Thus the Company making the fixed payments is actually paying

back less than what is has borrowed. So it is advantageous to it to go in for

the Fixed rate loan.


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Of course, the above calculations are based on perceptions regarding interest

rates. On a different stream of expected floating rates, the calculations would

have been different.

After a swap has been entered into, one of the parties will become a loser. If

the floating rates come down to a level lower than expected, the floating

payer will be better off and vice versa. Sometimes, the suffering counterparty

might want to unwind the swap and come out of its obligations for the rest of

the period. This can be done only with the consent of the other counterparty

and is likely to involve payment of the difference amount between the fixed

and floating present values of remaining streams, apart from whatever

penalty that is agreed upon. Alternatively, the suffering counterparty could

enter into another swap which when combined with the existing swap could

nullify the flows.

3.7 Collars mimicking swaps A costless collar will mimic a swap. Let us take a swap that involves

payment of a fixed rate of 8% interest on a given notional principal against

receiving the floating rate of interest. This is exactly equal to a collar,

whereby the holder has a cap struck at 8% and has written a floor with a

strike price of 8%. The payoff map is shown below:

Table V.2.4. Collars mimicking swaps Interest

rate Swap

in Swap out

Net Swap flow

Long Cap

Short floor

Net Collar

5% 5% 8% -3% 0 3% -3%

6% 6% 8% -2% 0 2% -2%

7% 7% 8% -1% 0 1% -1%

8% 8% 8% 0% 0 0 0%

9% 9% 8% 1% 1% 0% 1%

10% 10% 8% 2% 2% 0% 2%

:


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3.8 Summary Swaps are important Derivatives used in the management of interest rate

risks and currency risks. Like other Derivatives, swaps are also used by

speculators.

A plain vanilla interest rate swap involves the exchange of obligations by two

counterparties in such a way, that one which has to pay a fixed rate of

interest to its lending bank gets this reimbursed in part or full from the other

counterparty. Similarly, the second counterparty gets its floating rate due to

its bank reimbursed in part or full by the first counterparty. The exact

amounts to be exchanged and the periodicity are matters of contract and will

vary from case to case.

Comparative advantage is the basis of interest rate swaps. One of the

counterparties is likely to be lesser credit-rated and hence will have to pay

more interest on both fixed and floating segments. However, the amounts it

is required to pay in excess may not be the same. This gives rise to an

economic phenomenon called Spread Differential, which enables the two

counterparties to reap mutual benefits.

Currency swaps stem from the same principle, but arise out of multinational

obligations. This involves exchange of flows in two different currencies. The

effective result is that the borrowing entity deposits the amount it borrows in

exchange of getting a loan in the foreign currency. At the end of the contract

both the deposit and the loan are closed. Currency swaps will be useful only

if the amount to be paid as interest as a result of it is lower than what the

Company itself would have been able to get.


215

After a swap has been entered into changes in circumstances will make it less

or more valuable. Sometimes companies wish to come out of their swap

obligations by entering into another swap. Unwinding a swap may involve

paying a penalty and the difference in present values to the counterparty.

3.9 Key words

• Swap

• Plain-vanilla swap

• Comparative Advantage

• Spread Differential

• Currency Swap

• Unwinding of a swap

3.10 Questions for Self- study 1) How are the swap flows in a plain vanilla swap determined?

2) Is the exchange of principal an inherent assumption under Currency

swaps?

3) How do collars mimic swaps?

4) What is the relationship between Forward Rate Agreements and

Interest Rate Swaps?


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4 Credit Derivatives


• To understand the need for Credit Derivatives.

• To see the demonstration of some standard Credit Derivatives.

• To understand the economic function of Credit Derivatives.

4.2 Introduction In recent years, banking literature in India has been concentrating on

Operational Risk of Banks and the impact of the Basel II norms. While there

can be no doubt that this is a crucial feature of Bank management, the

development of Credit Derivatives which is becoming increasingly popular

abroad does not seem to have captured the imagination of the Indian Banker

to a great extent. This paper seeks to give the broad features of some common

derivative products and examines possible difficulties in their becoming

common in India. Admittedly, the paper looks more at issues involved than

clear solutions or possible guidelines to solve them. This is because the rules

by themselves are not an isolated piece and have to be integrated with other

policy initiatives of the Reserve Bank of India and the Government from time

to time.

The extant credit risk management practices involve monitoring and

constant follow-up on loan accounts. But cyclically, banks have the problem

of having to manage funds and risk concurrently. Credit Derivatives seek to

transfer the returns and risk of an asset portfolio without transferring the

ownership per se. They are thus off-Balance Sheet items. The basic idea is to

un-bundle the risk underlying an asset and trade it separately. The success

of these instruments will depend on a continuing market for various risk-


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denominated securities and a market that follows at least the semi-strong

Efficient Market Hypothesis. The fear of misuse and the whole question of

recourse and timing thereof have all slowed down the process of their

acceptance in India.

4.3 Common Credit Derivatives Types of Credit Derivatives permitted

The draft guidelines issued by the Working Group on Credit Derivatives

constituted by the Reserve Bank of India, seek to divide Credit Derivatives

into two broad categories.

Category 1 – involving buying and selling of specific protection in respect of

credit

Category 2 – involving swaps of total returns on assets.

Under Category 1 have been listed Credit Default Swaps, Credit Default

Options, Credit Linked Notes, and Credit Linked Deposits. And

Collateralized Debt Obligations

The present set of rules does not permit commercial banks absolute freedom

in using Credit Derivatives as buyers and sellers. Only plain instruments of

the plain vanilla type are permitted and that too and cannot be trading

intent, except to a limited extent. The deals should be on the basis of market

rates and free availability of information. A full gamut of systems and

procedures must be in place before any Bank can embark upon this activity.

4.4 Credit default swap The Credit Default Swap involves an arrangement with a counterparty by

which the later assumes the risk of the specified underlying asset for various

types of default (called credit events). In consideration for this, the

originating Bank pays a premium. On the happening of the credit event, the


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Bank has a claim on the counterparty and the latter steps into the shoes of

the former for trying to recover the dues from the default.

There are several issues involved in the setting up of a credit default swap

agreement. Specifically the following points need attention:

• Notional Value – The value of the assets under question and how their

final values will be determined

• Maturity Value- The methodology followed for ascertaining the

notional value can be followed for maturity value too

• Premium – the extent of premium to be paid for the agreement and the

mode and timing of payment are to be specified

• Definition of credit event – the event which would unambiguously

trigger the payment of the agreed amount. Preferably the credit event

should be one that is transparent and capable of being verified

• Extent of compensation in case of default – the extent of coverage for

default should be spelt out. This could vary among various credit

events

• Mode of settlement – The agreement will have to spell out the mode of

settlement and the timing of payment to be effected

• How the collaterals are treated after the payment – the counterparty

steps into the shoes of the Bank and will have the right to enforce the

collaterals. The procedure for transfer of this right and the timing

thereof has to be spelt out.

• Reference entity for settling disputes – It would be a good idea to fix an

independent third party to whom disputes could be referred for speedy

redressal.


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4.5 Total Return Swap The Total Return Swap involves not only the passing on of the default risk,

but also the assignment of the market risk as well. Here, the Bank will

undertake to pass on periodic returns on a specified set of assets to

counterparty. The return would not only involve the revenue returns of

interest, but also appreciation/depreciation in market value. The inherent

assumption under this is that the underlying assets are traded and have a

ready market. The counterparty takes the risk of the revenue returns being

not there and a negative market return. But potentially, the counterparty

can earn the full returns and any appreciation in the assets. As a

consideration, the counterparty pays a floating rate of return to the Bank.

The motivation of the bank could be that some of its heavy debts should be

changed and converted into more certain inflows. The counterparty might

want to take a little more risk and wishes to exchange relatively low interest

rates for potentially higher bond returns.

4.6 Collateralized Debt Obligations ( CDOs) The Collateralized Debt Obligations (CDOs) are a way of bundling credit and

selling it off to various interested counterparties depending upon their risk-

return balance. For this, the bundle of debt instruments (loans or bonds with

varying tenures, returns and perceived risks) are bundled together and

assigned to a Special Purpose Vehicle (SPV) formed for the purpose. The

SPV in turn informs to interested counterparties about the existence of this

bundle. The bundle itself is then split into suitable tranches of differing risks

and returns. For instance, the first 5% of default (highest risk) of the

portfolio can be labeled as Tranche 1 and can carry a very high return. The

investor segment subscribing to this tranche will be willing to assume the

higher risk for the higher expected return. This will be akin to subscribing to


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a junk bond. The tranches after this will have reduced levels of expected risk

and correspondingly reduced levels of expected return. The last tranche can

even be less risky than the original portfolio (since the expected level of risk

has been taken over by earlier tranches) and need only be rewarded with low

return. This will be similar to investing in an AAA bond. For the originating

bank, the loan has effectively been collected, while for the counterparty this

is an avenue of investment based on risk aversion. The SPV is merely a

facilitator.

The regulator might insist on basis capital adequacy norms to ensure no

default of the participants to the credit derivative transaction. These

stipulations, however, make it difficult for a deal to become operational.

4.7 An example of CDO A Bank has a portfolio of loans worth Rs.100 million on which there are some

chances of default. The portfolio currently yields 9%. The Bank wants to

come out of this holding. Through an intermediary a Special Purpose Vehicle

(SPV) is launched and the asset is transferred to it. The SPV then divides

this portfolio into various tranches of risk. For instance scheme like the one

shown below could be drawn up:

Table V.4.1. CDO demonstration

Tranche no. Amount Level of risk

from default Yield

1 Rs.20 mn. First 10% 16%

2 Rs.15 mn. Next 15 % 11%

3 Rs.25 mn Next 15% 9%

4 Rs.40 mn. Rest 4.75%


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Under this the subscribers to the first tranche get a higher return but are

exposed to great risk. The next tranche takes some risk, but lesser than the

first tranche and hence is rewarded only with a lower return and so on. The

last tranche has virtually no risk, since the first 40% of risk of default will be

borne by the other tranches. This tranche thus becomes as good as a AAA

bond, and enjoys the lowest return. It may be noted that the weighted

average of the returns to all the four tranches comes to 9%, which the yield of

the portfolio in the first place.

4.8 The Indian scenario In the Indian scenario a number of Banks would have situations that can

safely warrant a Credit Default Swap. The motivation of the counterparty

has to be based on core competence to monitor outstanding,, the need to have

this portfolio in their Balance Sheet to augment income, and to optimize the

overall risk exposure. The last point is subject to the regulatory authority

specifying capital adequacy and liquidity norms for the maximum extent of

such exposure.

The Total Return Swap is fraught with more regulatory problems. Here the

Bank seeks to transfer the entire return on the debt portfolio in return for a

floating rate yield. The Bank’s motivation is again to get rid of the risk

involved and pay the price for it by accepting a lower return. The

counterparty’s motivation is to have an avenue for investment with moderate

risk and corresponding high return. The problems arise as to what securities

the first Bank can be allowed to assign. In particular the following points

merit consideration:

1. Definition of Total return. This involves the market yield and the

tradability or otherwise of the underlying. Also, broad norms for

categorizing various investments in terms of Duration and Tenure may

be needed to make the volatility recognizable


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2. The counterparty takes the market risk as well, so it has to be clear as

to how the market rate is recognized and how frequently. In India,

debt instruments have not started trading in great volumes and so

price finding may be a task in itself.

3. The benchmark floating rate that the counterparty pays from time to

time to the originating Bank. The frequency of rate determination and

the timing of payments have to be clarified

4. Penalties for delay

5. Termination of the contract

6. Arbitration

Lastly, Collateralized Debt Obligations (CDOs) require a number of issues to

be clarified and regulated if it is to be done on a regular and large-scale basis.

Some of the broad issues involved are highlighted hereunder:

1. The bundling of the instruments to be transferred has to be done under

broad rules as to asset standard, maturity, and nature of securities.

Ultimately the SPV works under the principle of an ascertainable yield

for the whole basket split into separate yields commensurate with risk

taken by each tranche.

2. Classification of the assets in the bundle with criteria as to yield to

maturity, duration and tenure has to be clear

3. Norms as to the formation of Special Purpose Vehicles (SPVs) have to

clarify as to fixation of capital adequacy, management charges and

reporting.

4. One of the contentious issues in CDO is the fixation of the risk-return

balance for the tranches. Norms may have to go into the maximum

number of tranches, the range of yield differences in the tranches, and

some measure of “Beta” for the respective risk taken by each tranche,

so that potential participants can assess the position fast.


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5. The periodicity of payment of returns, assignment of the right to

receive the returns to the SPV, the passing on of respective tranche

returns, and maintenance of accounts by the SPV to satisfaction of the

participants and the regulator, all need attention.

6. Lastly, the timely resolution of disputes by arbitration should be

provided for in the rules.

4.9 Other aspects For these instruments to gain popularity more frequent deals involving these

should take place in the market and Banks should have perceivable benefits

from these. For this, we need a general market that is aware of the nuances

and willing to go ahead with a novel instrument with unknown risks. The

Regulator has a crucial role in bringing about awareness and developing a

measure of safety in deals. At first, it is likely that the regulation is very

stringent resulting in high capital adequacy, and risk coverage, but over a

period of time with experience of this deal, these are likely to become very

popular.

Apart from the general uses for credit risk management, Credit Derivatives

can also develop into a useful tool for Asset Liability Management (ALM) of

Banks. Banks who follow strategies based on Gap analysis will be concerned

about certain Rate Sensitive Assets or Liabilities and one of the additional

tools in the hands of the Bank would be to enter into a credit derivative

transaction.

4.10 Summary The Credit Derivative Market in India is in a nascent stage. With the full

development of the fixed income securities market and the advent of interest

rate Derivatives, the logical next step is the popularity of Credit Derivatives.


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The advantages of having a vibrant credit derivative market will be felt by all

sections of the market. The world over, innovations are being made to this

broad type of derivative and it is only a matter of time before we have this

working in full swing in India. Before that, the regulatory authorities will

need to have clear guidelines in place for the Banks and participating

Institutions. These guidelines will need to be flexible while taking care of

possible misuse. That is indeed a challenge.

The total returns swap and credit default swap are two common instruments

of Credit Derivatives. They seek to transfer the returns from the risky asset

to an interested buyer. Collateralized Debt Obligations create sub-bundles

from a pool of risky assets. The idea is to have a special purpose vehicle take

over the risky asset and then re -issue it as several instruments of varied risk

and return. Credit Derivatives are slowly gaining popularity and acceptance

in India..

4.11 Key words

• Credit Derivatives

• Total returns swap

• Collateralized debt obligations

• Credit default swap

• Special purpose vehicle

4.12 Questions for Self- study 1) Does the buyer of the risky security in a total swap get a higher return

on default not occurring?

2) In a CDO what is the role of the special purpose vehicle?

3) In a credit default swap what are common trigger events?


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5 Risk Management with Derivatives


• To understand the basic steps in risk management using Options

Greeks

• To understand the variability of portfolio value as a result of changes

in Delta and Gamma

• To understand the application of Delta Gamma hedging

• To have an overview of the hedging process in the Corporate Sector

5.2 Introduction Derivatives have a great deal of use in Risk Management. A judicial use of

Derivatives in the right proportion enables a Corporate manger to optimize

his risk-return matrix.

Basic hedging has already been discussed at the appropriate places in earlier

Chapters. Here we look at some sophisticated use of Option Greeks to

manage risk better.

One inherent assumption under most of the models given below should be

borne in mind. As we have seen, Options Pricing is a complex subject and

there cannot be any unanimity as to what factors influence the prices more.

Besides, the Black-Scholes model need not always reflect the correct price,

although the model is in great use and arguably the best suited. However, in

the examples and strategies illustrated below, we have taken the calculations

as per the Black-Scholes model for computational purposes. If the markets

consistently ignore the Black-Scholes model and go by another framework


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(there is no evidence of this), then we need to re-compute our projections on

that basis.

Option Greeks change in value with changes in parameters and so require

frequent watching. So any strategy that uses the values of Option Greeks

needs to be reviewed frequently.

Lastly, hedging is a continuous process involving planning and execution.

Derivatives offer an important additional tool in the hands of the Corporate

Finance Head to manage risk. However, most derivatives have some minus

points as well and these have to be balanced by the Company..

5.3 Hedging using Greeks

Position Delta

Position Delta refers to the Weighted average Delta of a portfolio of holdings

(of stock and Options) on the same underlying. Position Delta measures the

extent to which a portfolio changes for a small change in stock prices. This

can be understood easily with an example.

Table V.5.1. Data on stock and delta (Price in Rs.)

Instrument Quantity Price Delta

Stock 100 38 1

Call -50 1.34 0.42

Put 50 3.30 -.66

Portfolio Value

(Weighted Average)

3898

A portfolio manager has 100 shares long of a stock (present market price

Rs.38), and has written 50 calls on it with an Exercise Price of 40, and has


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bought 50 puts on the same stock with an Exercise Price of Rs.41. Let us

assume the volatility to be 25%, the time to go to be 3 months (0.25) and the

risk free interest rate to be 7%. Inputting the above data into the Black-

Scholes calculator we get the following:

The position delta is ((1*100 )+ (0.42* -50) +(-0.66 *50) = 46. This signifies

that an increase of the stock price by Re.1 will result in the portfolio value

going up by Rs.46. This can be verified as follows.

Let us assume the stock price goes up to Rs.39 the very next day. The

relevant figures then will be as follows:

Table V.5.2. Impact of change in stock price (Price in Rs.)

Instrument Quantity Price Delta

Stock 100 39 1

Call -50 1.80 0.50

Put 50 2.69 -.58

Portfolio Value

(Weighted Average)

3945

The actual value goes up by 47. But for rounding errors, the Position Delta

reflects changes reasonably accurately for small changes in the underlying.

After the change the Delta of the portfolio will also remain stable if the

Gamma is not substantial.

Apart from arbitrage operations, many market makers use this strategy to

lock up fluctuations arising out of stock movements, so that they can

concentrate on changes in bid-ask spreads


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Position Gamma

Position Gamma is the weighted average of Gammas of the various

components in the portfolio. We know that the Gamma of a call is equal to

the Gamma of a put and that it cannot be negative. However, a portfolio

gamma can be negative if it contains some short positions.

Let us take the case of a stock currently priced at Rs.80. A call on the stock

with 3 month maturity, 25% volatility, a risk free rate of return of 7% and

strike price of Rs.85 will have a price of 2.55, a Delta of 0.3888 and Gamma of

0.0383, as per the Black-Scholes model.

Let us take a Delta-hedged portfolio. It may be recalled that Delta Hedging

involves going short the calls and buying Delta times the stock. The position

is as follows:

Table V.5.3. Position Gamma – Price in Rs.

Instrument Quantity Price Delta Gamma

Call -100 2.55 0.388 0.0383

Stock 39 80 1 0

Portfolio Value 2865 0 -3.83

The Position Gamma of the portfolio is just the weighted average of the

Gammas of the components of the portfolio. We get (-100*0.0383) + (39*0)= -

3.83

Let us see the impact of small changes in the stock price on the very next

day.


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Table V.5.4. Impact of price changes on Gamma- Price in Rs.

Stock price

Call price

Call Delta

Call Gamma

82 3.4061 0.4661 0.0388

78 1.8498 0.3138 0.0364

When the price goes up to Rs.82, the portfolio value will be

(-100*3.4061) + (82 * 39) = 2857.

When the price comes down to Rs.78, the portfolio value will be

(-100*1.85) + (78 * 39) = 2857.

Either way the portfolio changes to a minimum extent but in the negative

direction. This happens whenever the Portfolio Gamma is negative. The

portfolio, of course, is delta hedged, so the value will not deteriorate by much.

Conversely, if the Portfolio Gamma had been positive ( as a result of a Delta

hedged portfolio consisting of long puts and long stock), the changed value

would still have been around the Delta-hedged level, but would have moved

slightly to the positive direction.

This is demonstrated below:

Let us take a delta-hedged portfolio consisting of long puts and long delta

times the stock. As in the last case, the stock is currently priced at Rs.80 and

we seek to go long to the extent of 100 puts at an Exercise Price of Rs.85.

There are 3 months to go in the contract the risk-free rate is 7%. And the

volatility is 25%.


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Table V.5.5. Delta hedging with puts – Position Gamma- Price in Rs.


Put +100 6.08 -0.611 0.0383

Stock 61 80 1 0

Portfolio Value 5488 0 3.83

The position gamma of the portfolio is (100*0.0383) + (61*0)= 3.83

Let us see the impact of small changes in the stock price on the very next

day.

Table V.5.6. Impact of price changes and Put Gamma- Price in Rs.

Stock price

Put price

Put Delta

Put Gamma

82 4.93 -.53 0.0388

78 7.37 -.69 0.0364

When the price goes up to Rs.82, the portfolio value will be (100*4.93) + (82 *

61) = 5495.

When the price comes down to Rs.78, the portfolio value will be

(100*7.37) + (78 * 61) = 5495.

Either way the portfolio changes by a small margin, but in the positive

direction.

If Gamma is small, Delta moves slowly and so rebalancing need not be very

frequent. Position Gamma gives an indication of how much the portfolio’s

Delta will fluctuate because of stock movements. This, in turn, gives an

indication of the re-balancing strategy to be followed.


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5.4 Delta-Gamma hedging Delta hedging works for small changes in stock prices. However, if the

portfolio manager wants to be more certain about the success of his strategy,

he must hedge the Gamma as well. The objective is to be not only Delta

hedged but also hedged for Gamma movements. Not only is the Delta of the

portfolio made 0, but the Gamma too is made 0, so that changes in prices of

stock will not affect changes in value of the portfolio (a result from Delta-

hedging), but will not substantially change the value of the Delta itself ( a

result from Gamma hedging). This ensures that the portfolio need not be

rebalanced from time to time.

An example will show the effect of this.

Let us take the following situation:

Table V.5.7. Delta-Gamma hedge – facts (Amount in Rs.)

Stock Price 120

Strike Price 125

Volatility 15%

Risk-free rate 8%

Time 3 months

A trader wants has sold 100 calls on the above strike price at a premium of

Rs.2.50. (This can be verified from the Black-Scholes calculator). She wants

to be Delta-Gamma hedged. The Delta of the call is 0.4051 and the Gamma

0.0431.

To be delta-hedged the Delta times the number of stock has to be bought.

This entails buying of 41 shares at the current market price of Rs.120. The

portfolio value is:


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Table V.5.7. Portfolio position for Delta-Gamma hedge- Amount in Rs.


Call -100 2.5 0.4051 0.0431

Stock 41 120 1 0

Portfolio Value 4670 0 -4.3

In order to be Gamma hedged as well, another call on the same stock has to

be used in conjunction with this to make the total Gamma 0 along with the

Delta of 0. Suppose there is another call going with a strike price of Rs.130.

Its call price as per the Black-Scholes calculator will be Rs.1.11, and its Delta

and Gamma will be 0.22 and 0.03 respectively.

Step 1 – Solve for Gamma of portfolio to be 0

We have to compute the position we have to take in the new Call (with strike

price Rs.130) in order to make the original portfolio of -100 call (strike price

Rs.125) and long 41 stock, Gamma neutral

0 = (0.0431*-100) + (0* 41) + (x * 0.03)

Solving we get that we need 143.66 (say 144) numbers of new calls (Exercise

Price Rs.130) to make the portfolio Gamma neutral

Step 2- Making the portfolio Delta- neutral

So the portfolio now consists of (-100) calls of Exercise Price Rs.125, and +144

calls of Exercise Price Rs.130) along with some stock. The number of stock to

be held has now to be adjusted in such a way that the total Delta equals 0.

0 = (0.4051*-100) + (1* S) + (144x * 0.22)

Solving we get S to be 8.83. We require 8.83 (say 9) shares to make the

portfolio Delta- neutral.

The Delta- Gamma-Neutral portfolio will consist of


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-100 calls (Exercise Price Rs.125)

+144 calls (Exercise Price Rs.130)

+9 shares

Let us verify our finding for some changes in stock prices and take two

scenarios by which stock prices go to Rs.122 and Rs.118.

Table V.5.8. Delta-Gamma hedge- verification

(Amount in Rs)

Scenario Stock Call (strike 125)

Call (strike 130)

Total

Base case position

9 -100 144

Rs.120 Rs.2.5 Rs.1.11 Rs.989.84 If stock price becomes 122

Rs.122 Rs.3.40 Rs.1.63 Rs.992.72

If stock price becomes Rs.118

Rs.118 Rs.1.78 Rs.0.73 Rs.989.12

The portfolio value remains at around the base-case level. All the figures are

taken using the Black-Scholes calculations.

5.5 Discussion on Hedging Policy A Company’s risk management policy will depend upon a number of factors.

At the outset, it is necessary to estimate the levels of expected return

that the Company expects from its various operations and the risk it is

being exposed to.

This involves a two-way analysis of the return and fluctuations to the return

likely. The word “risk” denotes the levels to which the expected returns can


234

fluctuate based on various scenarios. The firm has to take a calculated view

of the risk and the levels to which risk should be reduced. The following

points may be noted in this connection:

1. In an ideal market, higher expected returns can be got only by taking

correspondingly higher levels of risk. A Company has to evolve a

policy of the level of risk it wants to expose itself to and

correspondingly plan for its returns.

2. The risk levels could be analyzed from two specific angles – the

business-related risk and financial risk. Business-related risk is what

all companies will be exposed to because of factors of production,

governmental policy, environmental issues, tax exposure, competition

to the industry etc. Financial risk, on the other hand, is more micro in

nature and depends on the Company’s policy regarding use of debt

finance, its treasury operations and its reinvestment policy.

3. Operational risk arising out of business risk can be covered by having

suitable contractual terms and sometimes by buyback or abandonment

clauses.

4. Financial risks need to be covered with financial instruments.

Derivatives play a big part in this process.

5. These risks may arise out of uncertainty of financial parameters like

exchange rates or asset prices. These risks could then be covered by

using Futures or Options. The instruments to be used will have

different levels of cost and utility. Thus, Futures lock up prices at a

fairly low cost, while Options will give the Company both the

advantages, but at a higher cost.

6. In forming a risk management policy, the Company has to take into

account the cost-benefit angle. There could be several views on this. A

threshold Company might not want to take too much risk on its inflow

and might hedge all its inflows. An established high-growth Company

might decide to leave a little bit of its financial risk open so that it can


235

reap benefits from favorable movements. This is done with the

confidence that even if the conditions are adverse, the loss can be borne

because of the overall success of the Company. Some successful

companies have been known to remain conservative and continue to

manage risk with hedging, and not be tempted by possibilities of risky

super profits.

7. The extent of hedging required is also a matter of policy and financial

prudence. Hedging is basically an insurance. One may not always

need 100% coverage. Even if one wants it, instruments can guarantee

only substantial coverage and not always full converge. Within that

framework, Companies have been known to form specific policies

beforehand as to

a. The extent of hedge to be carried out for various types of

exposure

b. The instrument to be used for hedge

c. The review process of the hedge and rebalancing

d. The limit to cost of hedging

e. The goal from the hedging process

8. Value at Risk is a statistical computation which will enable the

Corporate manger to know the maximum extent to which the

portfolio’s value can deteriorate over a given period with a probability

of 95% or 99%. This figure (always in absolute amounts) gives a fairly

good indication of the maximum risk exposure that the Company has.

Accordingly a suitable risk management policy can be drawn up.

9. The use of Derivatives in unison or in combination offers a wonderful

opportunity for the Corporate Finance head. Derivatives in their

simple form have become very popular and over time the use of

sophisticated instruments is also likely to go up.

10. Derivatives have great use as hedging instruments in the stock

portfolio sector. One major criticism of Derivatives has been that the


236

hedge practice using these instruments is not always transparent.

Over time it is necessary to have regulatory systems within the

Company to make sure that errors of judgment do not occur.

Derivatives accounting is also a complex subject. Exposure

management and suitable hedging of excessive exposure is the key to

good portfolio management

5.6 Summary Option Greeks are widely used in risk management. Inherently, the corporate

hedger assumes that the prices and other parameters relating to the Options

will remain within predictable limits. The Black-Scholes model is used for

calculating the Greeks. For small changes in the underlying hedging policies

using Greeks are likely to succeed.

Position Delta refers to the average Delta of the portfolio. Position Delta will

tell us the extent to which the portfolio value is likely to go up or down given

a small change in the stock prices. Position Delta is likely to be very accurate

only for small changes in stock value.

Position Gamma relates to the average Gamma of the portfolio. Gamma

measures the extent to which Delta itself is likely to change with change in

stock prices. One of the great anxieties of a Delta hedger is the need to

frequently rebalance the portfolio. If the Gamma is low or close to 0, the need

to change the portfolio is not very frequent. A position Gamma gives the

measure of overall Gamma for a portfolio

A Delta Gamma hedge makes sure that the portfolio manager’s hedging

strategy works correctly and that Delta’s rebalancing frequency will not

greatly affect the portfolio’s hedge performance.


237

In framing a Corporate Hedging Policy, the extent of return desired, the risk

to be taken to meet the return expectation and the instruments to be used all

play a role. Though the use of Derivatives in a naked way is fraught with

risk, their use in Risk Management and Hedging make them powerful tools

of the future.

5.7 Key words

• Position Delta

• Position Gamma

• Delta Gamma Hedge

• Hedging

• Value at Risk

5.8 Questions for Self- study 1) How is Position Gamma calculated? Can it be negative?

2) What does a positive Position Gamma show?

3) What is the principle behind the Delta-Gamma hedge?

4) If prices do not conform to the Black-Scholes model, will the above use

of Greeks still work?

5) What factors must be considered by the Corporate Fund Manager in

drawing up a hedging policy?


238

References

1. Ahn Mark J Strategic Risk Management Mcgraw-Hill, 1991 2. Apte P G International Financial Management Tata McGraw-Hill,

2003 3. Arak Marcelle, Aruro Estrella, Laurie Goodman and Andrew Silver.

Interest Rate Swaps: An alternative explanation, Financial Management 17 (1988):12-18

4. Beder, Tanya Styblo. Equity Derivatives for Investors, The Journal of Financial Engineering 1 (September, 1992): 174-195

5. Bernstein, Peter L. Against the Gods: The Remarkable Story of Risk. New York. Wiley, 1996

6. Bicksler, James and Andrew Chen. An Economic Analysis of Interest Rate Swaps. The Journal of Finance 41 (July 1976): 645-656

7. Black, Fischer. “How to Use the Holes in Black-Scholes” Journal of Applied Corporate Finance 1 (Winter 1989): 67-73

8. Brennan, Michael J. “The Pricing of Contingent Claims in Discrete Time Models.” The Journal of Finance 34 (1979): 53-68

9. Chance Don M. “An Introduction to Derivatives and Risk Management”. Harcourt 5th edition (2006)

10. Bender Ruth Corporate Financial Strategy Oxford , 2003 11. Benhorim Moshe Essentials of Corporate Finance Bacon Inc., 1987 12. Berclay Michael J and Smith Clifford W Jr. The Maturity Structure of

Corporate Debt Journal of Finance, June, 1995 13. Bodie, Kane and Marcus Investments Tata Mcgraw Hill, 2002 14. Brealey Richard and Myers Stewart C Principles of Corporate Finance

Tata Mcgraw Hill, 2003 15. Brigham Eugene and Ehrhardt Mike Financial Management: Theory

and Practice South Western , 2002 16. Butler Kirt C Multinational Finance South Western, 2000 17. Chandra Prasanna Financial Management: Theory and Practice Tata

Macgraw-Hill, 2005 18. Damodaran Aswath Damodaran on Valuation John Wiley,1994 19. Eiteman David K Multinational Business Finance Addison

Wesley,2001 20. Ellsworth Richard Subordinate financial policy to corporate strategy,

Harvard Business Review, December,1983 21. Emery Douglas Corporate Financial Management Prentice Hall,1997 22. Errunza Vihang R International Business Finance Global Business

Press,1994 23. Eun Cheol and Resnick Bruce International Financial Management,

McGraw Hill,2002


239

24. Fisher and Jordan Securities Analysis and Portfolio Management, Pearson Education,2003

25. Froot Kenneth A A framework for risk management, Harvard Business Review, November-December,1994

26. Levi, Maurice D International Finance, the markets and Financial Management of multinational Business McGraw Hill, 1990

27. Bagchi S K, Operational Risk Mitigation Guidelines – Scope for Refinement for Indian Banking, , Professional Banker, August, 2006

28. Bent Christensen and Prabhala Nagpurnanand The Dynamic Relationship between Implied Volatility and Realized Volatility, Journal of Finance, ,(1996) Jul,1996, Vol.51.Issue 3, pp 1033

29. Corrado Charles and Su Tie, Implied Volatility Skews and Stock Index Skew ness and Kurtosis implied by S&P 500 Index Option Prices, The European Journal of Finance (1997), Vo.3, March, 1997, pp. 73-85

30. David Dubofsky and Thomas, Derivatives- Valuation and Risk Management, Miller, Oxford University Press, 2003.

31. Dennis, Patrick; Mayhew, Stewart; Stivers, Chris. Stock Return, Implied volatility innovations and asymmetric volatility phenomenon, Journal of Financial & Quantitative Analysis (2006), 41(2), 381-406.

32. Eraker Bjorn, Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Options Prices, Journal of Finance, June, 2004. p.1367

33. Fleming Jeff, The Quality of market volatility forecasts implied by S&P 100 index option prices, Journal of Empirical Finance (1998) 5. p.2059

34. Gupta V K, Risk Management Systems in banks, The Indian Banker, May, 2006

35. Gwilym Owain Ap and Buckle Mike (1999), Volatility forecasting in the framework of the option expiry cycle, the European Journal of Finance, Vol. 5, March, 1999., Vo.5, pp.73-94

36. Haeberle Carlos G and Kahl Kandice H, , A Comparison of Historic and Implied Volatility for Predicting Agricultural Option Premiums, Atlantic Economic Journal, (1991), Mar.1991.pp.39-88

37. Hull C John, Options, Futures and Other Derivatives, Prentice Hall (2003) 5th edition.

38. Kumar SSS, Financial Derivatives, Prentice Hall, 1st edition (2007) 39. Madtha CM, Durba Roy and Kasmera Dilip Singh, Risk Management

in Indian Banking Industry in 2015, , Professional Banker, July, 2006 40. Rakesh Mohan, Recent Trends in Indian Debt Market and Current

Initiatives, RBI Bulletin, April, 2006 41. Rene M Stulz, Risk Management and Derivatives, South Western 42. Robert A Strong, Derivatives – An Introduction, South-Western, 1st

edition, 2002.


240

43. Robert W. Kolb, Futures, Options, and Swaps, Blackwell Publishing, 4th edition, 2003.

44. Rubinstein, Mark, Alternative paths to portfolio insurance, Financial Analysis Journal (1985), 41(4), 42-52.

45. Saikrishna K, Commercial Banks in India- Challenges ahead, Professional Banker, September, 2006

46. Scott H Fullman, Options a personal seminar, New York Institute of Finance, 1992.

47. Schewert William G, Stock Volatility in the New Millennium, How Whacky is NASDAQ, Working Paper, national Bureau of Economic Research, w/8436, (2001), www. nber.org/papers/w8436

48. Schonbucher Philip, A Market Model for stochastic implied volatility, Mathematics of Finance, August,1999, pp.2071-2092

49. Stulz M Rene, Risk Management & Derivatives (2003), South Western, 1st edition.

50. Whaley, and Robert, Valuation of American call options on dividend paying stocks: Empirical tests, Journal of Financial Economics, (1982)10, 29-58.

51. www.nseindia.com 52. Draft Guidelines for Introduction of Credit Derivatives in India,

DBOD-BP-1057 dated 26/03/2003 53. Anderson David, Sweeney Dennis, Williams Thomas Quantitative

Methods for Business Thomson , 1983


241

INDEX American Option, 4, 73, 75, 77, 78, 104,

108, 109, 110, 111, 113, 116, 117, 119, 151, 152

At-the money, 73 Basis, 2, 34, 36, 40 Bear Spread, 131, 132, 134, 139 Beta, 3, 38, 61, 62, 63, 64, 67, 68, 176,

222 Binomial model, 1, 5, 141, 153, 161,

162, 164, 172 Black-Scholes model, 1, 5, 163, 164,

165, 169, 172, 173, 176, 183, 225, 228, 236, 237

Bounds, 111 Box Spread, 137, 138, 139 Bull Spread, 128, 129, 130, 139 Butterfly spread, 4, 134, 135, 139, 140 Call, 4, 78, 80, 82, 87, 91, 94, 95, 96,

100, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 123, 124, 126, 127, 128, 131, 132, 135, 137, 138, 142, 143, 144, 145, 146, 147, 151, 165, 169, 178, 183, 186, 201, 226, 227, 228, 229, 232, 233

Cap, 199, 213 Capital Assets Pricing Model, 47,

174 Collar, 201, 213 Collateralized Debt Obligations, 6, 217,

219, 222, 224 Cost of carry, 2, 23, 27, 44, 51, 52 Covered call, 96, 103 Covered call writing, 103 Covered Call writing, 3, 91, 95, 103 Credit Default Swaps, 217 Currency Swap, 6, 209, 210, 215 Delta, 1, 5, 7, 181, 182, 183, 184, 185,

186, 187, 188, 189, 190, 225, 226, 227, 228, 229, 230, 231, 232, 233, 236, 237

Delta Hedging, 1, 5, 181, 185, 190, 228 Delta-Gamma hedge, 231, 232, 233,

237 Discount yield, 192

Dividend, 42, 61, 110, 116 Dividends, 111 Early Exercise, 111 Eurodollar Futures, 191, 195, 201,

202 European Option, 73, 74, 77, 78, 111,

147, 164, 182 Exercise Price, 71, 72, 74, 75, 77, 81,

84, 87, 91, 92, 93, 94, 96, 106, 107, 109, 110, 111, 122, 123, 125, 128, 131, 133, 134, 136, 141, 142, 143, 144, 145, 147, 148, 149, 150, 151, 152, 153, 154, 157, 166, 167, 169, 170, 183, 226, 229, 232, 233

Floor, 200 Forward, 6, 10, 13, 15, 17, 18, 19, 21,

23, 24, 25, 26, 27, 28, 29, 30, 31, 34, 37, 41, 52, 53, 68, 191, 195, 201, 202, 203, 215

FRA, 195, 197, 198, 199, 200, 202 Futures, 1, 2, 3, 6, 8, 10, 14, 15, 17, 19,

21, 22, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 75, 76, 77, 78, 97, 98, 99, 102, 103, 191, 192, 193, 194, 195, 201, 202, 234, 239, 240

Gamma, 5, 7, 187, 188, 189, 190, 225, 227, 228, 229, 230, 231, 232, 236, 237

Hedging, 1, 2, 3, 6, 7, 28, 29, 39, 40, 57, 61, 67, 181, 183, 185, 190, 193, 226, 228, 233, 235, 237

In the money, 73 Index Futures, 1, 3, 54, 56, 57, 60, 62,

63, 67 Log normal distribution, 172 Margins, 32 Mark to market, 32 Mimicking portfolio, 103 No-arbitrage condition, 111 Optimal Hedge Ratio, 3, 61, 67, 68, 154,

161


242

Options, 1, 3, 4, 5, 10, 11, 14, 15, 70, 71, 73, 74, 75, 76, 77, 78, 79, 86, 87, 89, 91, 100, 102, 104, 105, 108, 109, 110, 111, 113, 116, 117, 119, 122, 141, 142, 147, 149, 151, 152, 153, 154, 158, 160, 161, 164, 168, 169, 171, 174, 176, 178, 181, 182, 189, 202, 217, 225, 226, 234, 236, 239, 240

Position Delta, 226, 227, 236, 237 Position Gamma, 228, 230, 236, 237 Protective puts, 103 Put, 1, 3, 4, 78, 83, 85, 87, 91, 97, 98,

99, 100, 101, 102, 103, 114, 117, 123, 124, 125, 126, 127, 129, 130, 134, 137, 138, 141, 143, 144, 145, 146, 147, 151, 178, 183, 188, 201, 226, 227, 230

Put-call parity, 146 Random walk, 172

Spread Differential, 206, 208, 214, 215

Stock Exchange, 55, 57, 75, 192 Straddle, 4, 122, 123, 124, 139 Strangle, 4, 125, 126, 139 Strike price, 73, 80, 82, 83, 84, 85, 87,

107, 109, 110, 114, 115, 116, 117, 118, 168

Swap, 6, 14, 204, 207, 210, 212, 213, 215, 217, 219, 221

Synthetic portfolio, 103 T-Bill, 193, 201, 202 Tenure, 110, 170, 221 Theta, 5, 187, 188, 189, 190 Value at Risk, 235, 237 Vega, 5, 187, 188, 189, 190 Volatility, 1, 5, 168, 171, 172, 173, 174,

179, 184, 231, 239, 240 YTM, 191

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Fundamentals of financial derivatives nrp

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