Financial Engineering chapter 01

transcript

Chapter 1. Introduction

Rangarajan K. Sundaram

Stern School of Business New York University

Outline

Introduction

Forward Contracts

Futures Contracts

Options

Derivatives and Risk-Management: Some Comments

Appendix: Interest-Rate Conventions

Introduction

ObjectivesThis segment

Introduces the major classes of derivative securities

ForwardsFuturesOptionsSwaps

Discusses their broad characteristics and points of distinction.

Discusses their uses at a general level.The objective is introductory: to lay the

foundations for the detailed analysis of derivative securities.

Derivatives

A derivative security is a financial security whose value depends on (or derives from) other, more fundamental, underlying financial variables such as the price of a stock, an interest rate, an index level, a commodity price or an exchange rate.

There are three basic classes of derivative securities:Futures & forwards.Swaps.Options.

Basic Distinctions – I Forward contracts are those where two parties

agree to a specified trade at a specified point in the future.

Defining characteristic: Both parties commit to taking part in the trade or exchange specified in the contract.

Futures and swaps are variants on the theme: Futures contracts are forward contracts where

buyers and sellers trade through an exchange rather than bilaterally.

Swaps are a kin to forward contracts in which the parties commit to a series of exchanges at several dates in the future.

Basic Distinctions – II Options: Characterized by optionality concerning the

specified trade. One party, the option holder, retains the right to enforce

or opt out of the trade. The other party, the option writer, has a contingent

obligation to take part in the trade. Call option: Option holder has the right, but not the

obligation, to buy the underlying asset at the price specified in the contract.

Option writer has a contingent obligation to participate in the specified trade as the seller.

Put option: Holder has the right, but not the obligation, to sell the underlying asset at the price specified in the contract.

Option writer has a contingent obligation to participate in the specified trade as the buyer.

Derivatives are BIG Business ... BIS estimates of market size (in trillions of USD):

... and a Rapidly Growing One BIS estimates of market size (in trillions of USD):

Risk-Management Roles - I These classes of derivatives serve important, but

different, purposes.Futures, forwards and swaps enable investors to

lock in cash flows from future transactions.Thus, they are instruments for hedging risk.

"Hedging" is the offsetting of an existing cash-flow risk.

Example 1 A company that needs to procure crude oil in one month can use a one-month crude oil futures contract to lock in a price for the oil.

Example 2 A company that has borrowed at floating interest rates and wishes to lock in fixed interest rate payments instead can enter into a swap where it commits to exchanging fixed interest rate payments for floating ones.

Risk-Management Roles - II Options provide one-sided protection. The option confers a right without an obligation. As a

consequence: Call Protection against price increase. Put Protection against price decrease.

Example Suppose a company needs to procure oil in one month. If the company buys a call option, it has the right to

buy oil at the "strike price" specified in the contract. If the price of oil in one month is lower than the strike

price, the company can opt out of the contract. Thus, the company can take advantage of price

decreases but is protected against price increases. In short, options provide financial insurance.

Outline for Remaining Discussion

The rest of the material defines these classes of instruments more formally.

Order of coverage:ForwardsFuturesOptionsSwaps

Forward Contracts

Forward Contracts The building block of most other derivatives,

forwards are thousands of years old. A forward contract is a bilateral agreement

between two counterparties a buyer (or "long position"), and a seller (or "short position")

to trade in a specified quantity of a specified good (the "underlying") at a specified price (the "delivery price") on a specified date (the "maturity date") in the

future. The delivery price is related to, but not quite the

same thing as, the "forward price." The forward price will be defined shortly.

Forwards: Characteristics

Bilateral contract Negotiated directly by seller and buyer.

Customizable Terms of the contract can be "tailored."

Credit Risk There is possible default risk for both parties.

Unilateral Reversal Neither party can unilaterally transfer its obligations in the contract to a third party.

Futures & forwards differ on precisely these characteristics as we see shortly.

Important characteristics of a forward contract:

The Role of Forwards: Hedging Forwards enable buyers and sellers to lock-in a price

for a future market transaction. Thus, they address a basic economic need:

hedging. Demand for such hedging arises everywhere.

Examples: Currency forwards: lock-in an exchange rate for a

future transaction to eliminate exchange-rate risk. Interest-rate forwards (a.k.a. forward-rate

agreements): lock-in an interest rate today for a future borrowing/investment to eliminate interest-rate risk.

Commodity forwards: lock-in a price for a future sale or purchase of commodity to eliminate commodity price risk.

BUT...

An obvious but important point: The elimination of cash flow uncertainty using a forward does not come "for free."

A forward contract involves a trade at a price that may be "off-market," i.e., that may differ from the actual spot price of the underlying at maturity.

Depending on whether the agreed-upon delivery price is higher or lower than the spot price at maturity, one party will gain and the other party lose from the transaction.

An Example A US-based exporter anticipates €200 million of

exports and hedges against fluctuations in the exchange rate by selling €200 million forward at $1.30/€.

Benefit? Cash-flow certainty: receipts in $ are known.

Cost? Exchange-rate fluctuations may lead to ex-post regret.

Exchange rate at maturity is $1.40/€. Exporter loses $0.10/€ for a total loss of

$20 million on the hedging strategy. Exchange rate at maturity is $1.20/€.

Exporter gains $0.10/€ for a total gain of $20 million on the hedging strategy.

Forward Contracts: Payoffs Forward to buy XYZ stock at F = 100 at date T. Let ST denote the price of XYZ on date T.

Forwards are "Linear" Derivatives

ST : Spot price at maturity of forward contract.F : Delivery price locked-in on forward contract.

The Forward Price

We have seen what is meant by the delivery price in a forward contract. What is meant by a forward price? The forward price is a breakeven delivery price: it is the delivery price that would make the contract have zero value to both parties at inception. Intuitively, it is the price at which neither party would be willing to pay anything to enter into the contract.

The Forward Price and the Delivery Price

At inception of the contract, the delivery price is set equal to the forward price. Thus, at inception, the forward price and delivery

price are the same. As time moves on, the forward price will typically

change, but the delivery price in a contract, of course, remains fixed.

So while a forward contract necessarily has zero value at inception, the value of the contract could become positive or negative as time moves on. That is, the locked-in delivery price may look

favorable or unfavorable compared to the forward price on a fresh contract with the same maturity.

The Forward Price Is the forward price a well-defined concept?

Not obvious, a priori. It is obvious that

If the delivery price is set too high relative to the spot, the contract will have positive value to the short (and negative value to the long).

If the delivery price is set too low relative to the spot, the situation is reversed.

But it is not obvious that there is only a single breakeven price. It appears plausible that two people with different information or outlooks about the market, or with different risk-aversion, can disagree on what is a breakeven price.

Futures Contracts

A futures contract is like a forward contract except that it is traded on an organized exchange.

This results in some important differences. In a futures contract: Buyers and sellers deal through the exchange, not directly. Contract terms are standardized. Default risk is borne by the exchange, and not by the

individual parties to the contract. "Margin accounts" (a.k.a "performance bonds") are used to

manage default risk. Either party can reverse its position at any time by closing

out its contract.

Forwards vs. Futures

The Futures Price

As with a forward contract, there is no up-front payment to enter into a futures contract. Thus, the futures price is defined in the same way as a forward price: it is the delivery price which results in the

contract having zero value to both parties. Futures and forward prices are very closely related but they are not quite identical. The relationship between these prices is examined in Chapter 3.

Options

Basic Definitions

An option is a financial security that gives the holder the right to buy or sell a specified quantity of a specified asset at a specified price on or before a specified date.

Buy = Call option. Sell = Put option On/before: American. Only on: European Specified price = Strike or exercise price Specified date = Maturity or expiration date Specified asset = "underlying" Buyer = holder = long position Seller = writer = short position

Broad Categories of Options

Exchange-traded options:Stocks (American).Futures (American).Indices (European & American)Currencies (European and American)

OTC options:Vanilla (standard calls/puts as defined above).Exotic (everything else—e.g., Asians, barriers).

Others (e.g., embedded options such as convertible bonds or callable bonds).

Options as Financial Insurance

As we have noted above, option provides financial insurance. The holder of the option has the right, but not the obligation, to

take part in the trade specified in the option. This right will be exercised only if it is in the holder's interest to

do so. This means the holder can profit, but cannot lose, from the

exercise decision.

Put Options as Insurance: Example Cisco stock is currently at $24.75. An investor plans to sell Cisco

stock she holds in a month's time, and is concerned that the price could fall over that period.

Buying a one-month put option on Cisco with a strike of K will provide her with insurance against the price falling below K.

For example, suppose she buys a one-month put with a strike ofK = 22.50.

If the price falls below $22.50, the put can be exercised and the stock sold for $22.50.

If the price increases beyond $22.50, the put can be allowed to lapse and the stock sold at the higher price.

In general, puts provide potential sellers of the underlying with insurance against declines in the underlying's price.

The higher the strike (or the longer the maturity), the greater the amount of insurance provided by the put.

Call Options as Insurance: Example

Apple stock is currently trading at $218. An investor is planning to buy the stock in a month's time, and is concerned that the price could rise sharply over that period.

Buying a one-month call on Apple with a strike of K protects the investor from an increase in Apple's price above K.

For example, suppose he buys a one-month call with a strike of K = 225.

If the price increases beyond $225, the call can be exercised and the stock purchased for $225.

If the price falls below $225, the option can be allowed to lapse and the stock purchased at the lower price.

In general, calls provide potential buyers of the underlying with protection against increases in the underlying's price.

The lower the strike (or the longer the maturity), the greater the amount of insurance provided by the call.

The Provider of this Insurance

The writer of the option provides this insurance to the holder: The writer is obligated to take part in the trade if the holder should so decide.

In exchange, writer receives a fee called the option price or the option premium. Chapters 9-16 are concerned with various aspects of the option premium including

the principal determinants of this price and models for identifying fair value of an option.

Chapter 17 discusses how to measure the risk in an option or a portfolio of options.

Chapters 18 and 19 extend the pricing analysis to "exotic" options. Chapter 21 studies hybrid securities such as convertible bonds that have

embedded optionalities.

What are Swaps?

A swap is a bilateral contract between two counterparties that calls for periodic exchanges of cash flows on specified dates and calculated using specified rules.

The contract specifies the dates (say, T1, T2, ... , Tn ) on which cash flows will be exchanged.

The contract also specifies the rules according to which the cash flows due from each counterparty on these dates are calculated.

The frequency of payments for the two counterparties need not be the same.

For example, one counterparty could be required to make semi-annual payments, while the other makes quarterly payments.

Categories of Swaps

Swaps are differentiated by the underlying markets to which payments on one or both legs are linked.

The largest chunk of the swaps market is occupied by interest-rate swaps, in which each leg of the swap is tied to a specific interest rate index.

Other important categories of swaps include: Currency swaps, in which the two legs of the swaps are

linked to payments in different currencies. Equity swaps, in which one leg (or both legs) of the

swap are linked to an equity price or equity index. Commodity swaps, in which one leg of the swap is

linked to a commodity price. Credit-risk linked swaps (especially credit-default

swaps) in which one leg of the swap is linked to occurrence of a credit event (e.g., default) on a specified reference entity.

What do Swaps Achieve?

Swaps are among the most versatile of financial instruments with new uses being discovered (invented?) almost every day.

One of the sources of swap utility comes from the fact that swaps enable converting exposure to one market to exposure to another market.

Example 1 Consider a 3-year equity swap in which One counterparty pays the returns on the S&P 500 on a

given notional principal P. The other counterparty pays a fixed rate of interest r on

the same principal P. The first counterparty in this swap is exchanging equity-market

returns for interest-rate returns over this three-year horizon. The second counter party is doing the opposite exchange.

Example 2 Consider an interest-rate swap in which One counterparty pays a floating interest-rate (e.g., Libor) on

a given notional principal P. The other counterparty pays a fixed rate of interest r on the

same principal P. Such a swap enables converting floating interest-rate exposure to

floating interest-rate exposure (and vice versa).

Example 3 Consider a currency swap in which One counterparty makes US dollar payments based on USD-

Libor. The other makes Japanese yen payments based on JPY-

Libor. The swap enables converting floating rate USD exposure to

floating-rate JPY exposure and vice versa.

Linking Different Markets

As a corollary, swaps provide a pricing link between different markets.

Consider the equity swap in Example 1. At inception, the fixed rate r in the equity swap is set so

that the swap has zero value to both parties, i.e., so that the PV of the cash flows expected from the equity leg is equal to the PV of the cash flows from the interest rate leg.

This means the interest rate r represents the market's "fair price" for converting equity returns into fixed-income returns.

Thus equity swaps also provide a pricing link between the equity and fixed-income markets: the swap not only enables transferring equity risk into interest-rate risk, it also specifies the price at which this transfer can be done.

Linking Different Markets

Similarly: Interest-rate swaps provide a link between different

interest-rate markets, for example, the fixed-rate at which floating-rate exposure can be converted to fixed-rate exposure.

Currency swaps provide a link between interest-rate markets indifferent currencies, for example, the EUR fixed rate at which USD floating-rate exposure can be converted to EUR fixed-rate exposure.

Swaps in this book

Part 3 of the book deals with swaps. Chapter 23 examines interest-rate swaps, the main component of the

market. Chapter 24 studies the characteristics, uses and pricing of equity swaps. Chapter 25 looks at currency swaps and commodity swaps. Other instruments with the "swaps" moniker are examined elsewhere in

the book. Variance and volatility swaps are discussed in Chapter 14. Credit-related swaps, including Total Return Swaps and Credit

Default Swaps are dealt with in Chapter 31.

Derivatives and Risk-Management: Some Comments

Derivatives and Risk-Management

Derivatives can be used to hedge or obtain insurance against existing risk exposures.

We examine derivatives use in various contexts throughout the book.

Here, we use a simple example to make a simple preliminary point: that derivatives do not offer a panacea in managing risk.

There are pros and cons to every derivatives strategy (including to the strategy of using no derivatives).

That is, there is no dominant alternative that is better in all conditions.

A Simple Example

Suppose it is currently December, and a US-based company learns that it will be receiving €25 million in March.

As a US-based organization, the company needs to convert the euros into dollars upon receipt, so is exposed to changes in the exchange rate.

The company has (at least) three choices: Do nothing, i.e., retain full exposure to changes in

exchange rates. Use a forward/futures contract to lock in an exchange

rate today. Buy a put option on the euro that guarantees a floor

exchange rate.

Comparing the Alternatives

We compare outcomes under these three alternatives using three relevant criteria:

1. Cash-flow uncertainty under the strategy.2. Up-front cost of the strategy.3. Exercise-time (or lock-in) regret from the strategy.

Assume the following: The company can lock in an exchange rate of $1.0328/€

using CME March futures contracts. The company can buy put options on the euro with a strike

of $1.03/€ and expiring in March at a total cost of $422,500.

The Alternatives Compared

The table below presents the outcomes (in US$) under the three alternatives in two scenarios:

A "low" exchange rate (relative to the locked-in rate) of $0.9928/€.

A "high" exchange rate of $1.0728/€.

The Alternatives Compared

1. Cash flow uncertainty Maximal for the do-nothing alternative. Intermediate for the option contract. Least for the futures contract.

2. Up-Front Cost Zero for the do-nothing and futures contract alternatives. Substantial ($422,500) for the options contract.

3. Exercise-Time Regret None with the options contract, but possible with the others:

If $1.0728/€. The futures contract has ex-post regret (not hedging would have been better).

If $0.9928/€. The do-nothing contract has ex-post regret: hedging would have been better.

The Best Alternative?

► There is none: Each strategy has its pros and cons.

Appendix: Interest-Rate Conventions

Interest-Rate Convention

One important preliminary issue is the interest-rate convention we use.

Any convention may be used—the choice is really one of convenience. Different interest-rate conventions are simply different mechanisms for converting sums of money due in the future into present values today, or investments made today into future values due at maturity.

As long as we obtain the correct present values and future values, it doesn't matter what convention we use.

Moving Between Conventions

Interest rates expressed under different conventions will not of course be the same (just as a person's height measured in inches is not the same as her height measured in centimeters).

But just as we can always convert height from centimeters to inches and vice versa, we can always move between the conventions and express a given situation in any interest rate convention that we want.

The key thing is to remember that an interest rate convention is simply a mechanism for telling us how to compute present values of future amounts (or future values of present investments).

Two Specific Conventions

For specificity, we use one of two conventions in the numerical illustrations:

Continuous-compounding. The money-market convention.

We discuss each below and also how to go from one convention to the other.

Remark The main body of the text uses mainly continuous-compounding. The money-market convention is introduced in the Exercises section in Chapter 3, and is used in several other chapters in the book.

Continuous–Compounding

The continuous-compounding convention is commonly used in theoretical work in modern Finance.

If the T-year continuously-compounded interest rate is r : $A invested for T years grows to $e rT A by time T. The present value of $A receivable at time T is PV (A) = e —rT A.

Continuous-compounding has several technical advantages which is why it is popular with modelers and is commonly used in finance textbooks.

Money-Market Convention The other convention we use is the money market convention. In the US money-market, an interest rate of ℓ over a horizon [0,T ]

means that the interest payable per dollar of principal is

where d is the actual number of days in the horizon [0,T ]. For example, if the 3-month interest rate is 5% and there are 91

calendar days in the 3-month horizon, then the interest received per dollar of investment is

Actual/360 is popular in other countries too, though some countries (such as Britain) use Actual/365.

Money-Market Convention

Under the Actual/360 convention, an amount A invested over [0,T ] at the rate ℓ grows by time T to

Conversely, the present value of A receivable at T is

Suppose an investment of $1 made today will be worth $1.03 in three months.

1. If the interest rate ℓ is expressed in the Actual/360 convention and the three-month horizon has 91 days in it, what is ℓ ?

2. If the interest rate r is expressed in the continuous-compounding convention and we treat three months as 1/4 years, what is r ?

Consider an investment of $1 over a horizon of one month. 1. If the interest rate ℓ expressed in the Actual/360 convention

is 4% and the one-month horizon has 31 days in it, to what does the invested amount grow to?

2. If you had to express the same outcome using a continuous-compounding convention, and we treat one month as 1/12 of a year, what is the continuously-compounded rate r ?

Consider an investment of $1 over a horizon of one month.

1. If the interest rate r expressed in the continuously–compounded terms is 4% and we treat the one month horizon as 1/12 of a year, to what does the invested amount grow?

2. If you had to express the same outcome using an Actual/360 convention and the one month horizon has 31 days in it, what is the rate ℓ ?

Financial Engineering chapter 01

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