CH3 Derivatives PP

7/29/2019 CH3 Derivatives PP

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Copyri ght 2011 by The McGraw-H il l Companies, Inc. All ri ghts reserved.McGraw-Hill/Irwin


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Chapter 3. Pricing Forwards and Futures I:The Basic Theory

Rangarajan K. Sundaram

Stern School of Business

New York University


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Outline

Introduction

Pricing Forwards by Replication

Numerical Examples

Currency Forwards

Stock Index Forwards

Valuing Forwards

Forward Pricing: Summary

Futures Pricing


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Introduction


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Objectives

This chapter introduces students to the pricing of derivatives using no-arbitrage considerations.

Key points:

1. The cost-of-carry method of pricing forward contracts.

2. The role of interest rates and holding costs/benefits in this process.

3. The valuation of existing forward contracts.


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The Forward Price

Forward price: The delivery price that makes the forward contract have

"zero value" to both parties.

How do we identify this zero-value price?

Combine

The Key Assumption: No arbitrage

with

The Guiding Principle: Replication.


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Pricing Forwards by Replication


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The Key Assumption: No Arbitrage

Maintained Assumption:Market does not permit arbitrage.

What is "arbitrage?"

A profit opportunity which guarantees net cash inflows with no net

cash outflows.

Such an opportunity represents an extreme form of market inefficiency where two identical securities (or baskets of securities)

trade at different prices.

Assumption isnot that arbitrage opportunities can never arise, but that

they cannotpersist.

This is a minimal market rationality condition: it is impossible to say

anything sensible about a market where such opportunities can persist.


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The Guiding Principle: Replication

Replication: Fundamental idea underlying the pricing ofall derivativesecurities.

The argument:

Derivative's payoff is determined by price of the underlying asset.

So, it "should" be possible to recreate (orreplicate) the

derivative's pay offs by directly using the spot asset and, perhaps,

cash.

By definition, the derivative and its replicating portfolio (should one

exist) are equivalent.

So, by no-arbitrage, they must have the same cost.

Thus, the cost of the derivative (its so-called "fair price") is just thecost of its replicating portfolio, i.e., the cost of manufacturing its

outcomes synthetically.

Key step: Identifying the replicating portfolio.


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Replicating Forward Contracts

Forward contracts are relatively easy to price by replication.

Consider an investor who wants to take along forward position.

Notation:

S: current spot price of asset.

T: maturity of forward contract (in years).

F: forward price for this contract (to be determined).

We will let 0 denote the current date, so Tis also the maturity date of forward

contract.


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Replicating Forwards

At maturity of the contract, the investor pays $F and receives one unit of

the underlying.

To replicate this final holding:

Buy one unit of the asset today and hold it to date T.

Both strategies result in the same final holding of one unit of the

underlying at T.

So, viewed from today, they must have the same cost.

What are these costs?


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Cost of Forward Strategy

The forward strategy involves a single cash outflow of the delivery price

Fat time T

So, cost of forward strategy: PV(F).


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Cost of Replicating Strategy

To replicate, we must

Buy the asset today at its current spot price S.

"Carry" the asset to date T. This involves:

Possible holding/carrying costs (storage, insurance).

Possible holding benefits (dividends, convenience yield).

Let

M= PV(Holding Costs)PV(Holding Benefits).

Net cost of replicating strategy: S+ M.


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The Forward Pricing Condition

By no-arbitrage, we obtain the fundamental forward pricing condition:

Solving this condition for F, we obtain the unique forward price consistent

with no-arbitrage.


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Violation of this Condition Arbitrage

IfPV(F) > S+ M, the forward isovervalued relative to spot.

Arbitrage profits may be made by selling forward and buying spot.

"Cash and carry" arbitrage.

Forward contract has positive value to the short, negative value tothe long.

IfPV(F) < S+ M, the forward isundervalued relative to spot.

Arbitrage profits can be made by buying forward and selling spot.

"Reverse Cash and carry" arbitrage.

The contract has positive value to the long and negative value to the

short.


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Determinants of the Forward Price

The forward price is completely determined by three inputs:

Current price S of the spot asset.

The cost M of "carrying" the spot asset to date T.

The level of interest rates which determine present values.

This is commonly referred to as the cost-of-carrymethod of pricing

forwards.

Two comments:

Forward and spot prices are tied together by arbitrage: they must

move in "lockstep."

To what extent then do (or can) forward prices embody expectations

of future spot prices?


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Pricing Formulae I: Continuous Compounding

Fundamental pricing equation: PV (F)= S+ M.

Let r be the continuously-compounded interest rate for horizon T.

So PV(F) = erTF.

Therefore, erTF= S+ M, so

When there are no holding costs or benefits (M= 0),


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Pricing Formulae II: Money-Market Convention

Let denote the T-year rate of interest in the money-market convention(Actual/360).

Ifd is the actual number of days in the horizon, then

Therefore, so


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Numerical Examples


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Example 1

Consider a forward contract on gold. Suppose that:

Spot price: S= $1, 140/ oz.

Contract length: T= 1 month = 1/12 years.

Interest rate: r= 2.80% (continuously compounded).

No holding costs or benefits.

Then, from the forward pricing formula, we have

Any other forward price leads to arbitrage.IfF> 1, 142.66, sell forward and buy spot.IfF< 1, 142.66, buy forward and sell spot.


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Abrbitrage with an Overpriced Forward

Suppose that F= 1, 160, i.e., it is overpriced by 17.34.

Arbitrage strategy:

1. Enter into short forward contract.

2. Buy one oz. of gold spot for $1,140.

3. Borrow $1,140 for one month at 2.80%.


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Abrbitrage with an Underpriced Forward

Suppose that F= 1, 125, i.e., it is underpriced by $17.66.

Arbitrage strategy:

1. Enter into long forward contract.

2. Short 1 oz. of gold.

3. Invest $1,140 for one month at 2.80%.


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Holding Costs and/or Benefits

Holding costs are often non-zero.

With equities or bonds, there are often holding benefits such as

dividends or coupons.

With commodities, there are often holding costs such as storage and

insurance.

Such interim cash flows affect the total cost of the replication strategy and

should be taken into account in pricing.

Our second example deals with such a situation.


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Example 2

Consider a forward contract on a bond.

Suppose that:

Spot price of bond: S= 95.

Contract length: T= 6 months.

Interest rate: r= 10% (continuously compounded) for all maturities.

Coupon of $5 will be paid to bond holders in 3 months.

What is the forward price of the bond?


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Example 2: The Forward Price

The coupon is a holdingbenefit. So M isminus the present value of $5 receivable in 3 months:

Thus, the arbitrage-free forward price Fmust satisfy

so

Any other forward price leads to arbitrage.


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Arbitrage with an Overvalued Forward

For example, suppose F= 98.

Then, the forward is overvalued relative to spot, so we want to sell forward,

buy spot, and borrow.

Buying and holding the spot asset leads to a cash outflow of 95 today, but

we receive a coupon of 5 in 3 months.

There are may ways to structure the arbitrage strategy. Here is one. We split

the initial borrowing of 95 into two parts, with

one part repaid in 3 months with the $5 coupon, and

the balance repaid in six months with the delivery price received on the

forward contract.


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The Arbitrage Strategy

So the full arbitrage strategy is:

Enter into short forward with the delivery price of 98.

Buy the bond for 95 and hold for 6 months.

Finance spot purchase by

borrowing 4.877 for 3 months at 10%

borrowing 90.123 for 6 months at 10%.

In 3 months:

receive coupon $5

repay the 3-month borrowing.

In 6 months: deliver bond on forward contract and receive $98

repay 6-month borrowing.


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Question: What is the arbitrage strategy ifF= 91.50?

Cash Flows from the Aribtrage


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Currency Forwards


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Forwards on Currencies & Related Assets

Forwards on currencies need a slightly modified argument.

For example, suppose you want to be long 1 on date T.

Two strategies:

Forward contract: Pay $F at time T, receive 1.

Replicating strategy: Buy x today and invest it to T, where x = PV (1).

PV(1) is the amount that when invested at the sterling interest rate will grow

to 1 by time T.

The "" inside the PV expression is to emphasize that present

values are being taken with respect to the interest rate.


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Currency Forwards: Replication Costs

Cost of the forward strategy in USD:

PV($ F) = F x PV($1).

Cost of the spot (or replicating) strategy in USD:Sx PV(1)

As usual, Sdenotes the spot price of the underlying in USD.

Here, the underlying is GBP, so Sis the spot exchange rate ( $ per).


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Currency Forwards: The General Pricing Expression

By no-arbitrage, we must have

Sx PV(1) = F x PV($1).

Solving we obtain the fundamental forward pricing expression for currencies:


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Currency Forwards with Continuous Compounding

Let rrepresent the T-year USD interest rate and d the T-year GBP

interest rate, both expressed in continuously-compounded terms.

Then, PV($ 1) = erTand PV(1) = edT .

Using these in the general currency forward pricing expression and

simplifying, we obtain


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Currency Forwards in the Money-Market Conventions

($): T-year USD interest rate (ACT/360).

(): T-year GBP interest rate (ACT/365). Then:

Substituting these into the general currency forward pricing expression

and simplifying, we obtain


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Example 3

Data:

Foreign currency: GBP

Spot exchange rate S(USD per GBP): 1.63146

Contract length T: 3 months = 90 days.

3-month USD Libor rate: ($) = 0.251% 3-month GBP Libor rate: () = 0.610%

Therefore, the unique arbitrage-free forward price is:

U d l d F d


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Undervalued Forward

Suppose we had F= $1.60/. Then, the forward isundervalued relative to spot, so we want to buy

forward, sell spot, and invest.

Long forward contract to buy 1 for $1.60 in 3 months.

Short PV(1). That is:

Borrow PV(1) = 0.9985 for 3 months at 0.61%.

Sell 0.9985 for $ at the spot rate of $1.63146/.

Invest the proceeds for 3 months at 0.251%.

In 3 months:

Pay USD 1.6000 and receive GBP 1 from the forward.

Repay GBP borrowing.


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Cash Flows from the Arbitrage

At inception:


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Cash Flows from the Arbitrage

At maturity:


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Currency Forwards: Exercise

Suppose you are given the following data (from 15-Jan-2010):

Spot USD/GBP exchange rate = $1.6347/.

Spot USD/EUR exchange rate = $1.4380/.

1-month Libor rates:

USD: 0.2331% (Actual/360) GBP: 0.5175% (Actual/365)

EUR: 0.3975% (Actual/360)

What are the 1-month USD/GBP and USD/EUR forward rates?

Actual 1-month forward rates on 15-Jan-2010:

USD/GBP: $1.62438/. USD/EUR: $1.43786/.


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We can also price forwards on stock indicesusing this approach.

A stock index is essentially a basket of a number of stocks.

If the stocks pay dividends at different times, we can approximate the

dividend payments well by assuming they are continuously paid.

Dividend yield on the index plays the role of the variable d in theformula.

Literally speaking, the idea of continuous dividends is an unrealistic one,

but, in general, the approximation works very well.

Computationally, much simpler than calculating cash value of dividend

payments expected over contract life and using the known-cash-payoutsformula.

l d d


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Example 4: Index Forwards

Data:

Current level of S&P index: 1,343

One-month interest rate (continuously-compounded): 2.80%

Dividend yield on the S&P 500: 1.30% What is the price of a one-month (= 1/12 year) futures contract?

In our notation: S= 1343, r= 2.80%, d= 1.30%, and T= 1/12.

So the theoretical futures price is

S&P 500 F t P i J 15 2010


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S&P 500 Futures Prices: Jan 15, 2010

Spot: 1136.03 (S&P on Jan 15, 2010)


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Valuing Forwards

Valuing Existing Forwards


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Valuing Existing Forwards

Consider a forward contract with delivery price K that was entered into earlierand now has T years left to maturity.

What is the current value of such a contract?

We answer this question for the long position.

The value of the contract to the shortposition is just the negative of the

value to the long position.

So suppose we are long the existing contract.

Suppose also that the current forward price for the same contract (same

underlying, same maturity date) is F.

Off tti th E i ti F d


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Offsetting the Existing Forward

Consider offsetting the existing long forward position with a short forward

position in a new forward contract.

Original portfolio:

Long forward contract with delivery price K and maturity T.

New portfolio: Long forward contract with delivery price K and maturity T.

Short forward contract with delivery price F and maturity T.

Value of original portfolio = Value of new portfolio (why?).

Val ation b Offset


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Valuation by Offset

What happens to the new portfolio at maturity?

Physical obligations in the underlying offset.

Net cash flow: FK.

So new portfolio - certainty cash flow ofFKat time T.

This means: Value of New Portfolio = PV(FK).

Therefore:

Value of Long Forward = PV(FK).

and

Value of Corresponding Short Forward = PV(KF).

Val ing Fo a ds S mma


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Valuing Forwards: Summary

Data:

Given forward contract: delivery price K, maturity date T.

Current forward price for same contract: F.

Valuations:

Value of long forward: PV(FK).

Value of short forward: PV(KF).

Intuition?

E l 5 V l i E i ti F d


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Example 5: Valuing Existing Forwards

You enter into a forward contract to sell 10,000 shares of Dell stock in

3months' time at a delivery price of $25.25.

A month later:

The price of Dell is $25.40. The two-month rate of interest at thispoint is 4.80% (money-market convention). There are 61 days in the

two-month period.

Dell is not expected to pay any dividends over the next two months.

What is the value of the contract you hold?

Example 5: The Steps Involved


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Example 5: The Steps Involved

To answer this question, we proceed in two steps:

1. Identify the forward price F today for delivery in two months.2. Use F together with the locked-in price K= 25.25 to identify the value of

the forward contract.



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Since there are no dividends, the forward price must satisfy PV(F) = S.

This means

or

Example 5: The Contract Value


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Example 5: The Contract Value

Since you are short the forward, the value of the forward contract is

PV(K F) = PV(0.36) = PV(0.36).

Using the 2-month interest rate of 4.80%, this present value is

Over 10,000 shares, therefore, the value of the contract is

10,000 0.3571 = 3,571


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A forward contract is acommitment by buyer and seller to take part in a

fully specified future trade.

The commitment to the trade makes forward payoffslinear.

Theforward price is that delivery price that would make the contract have

zero value to both parties at inception.

The forward price can be determined by replication, and depends on the cost

of buying and "carrying" spot.

Thevalue of a forward contract is the present value of the difference

between the locked-in delivery price on a contract and the current forward

price for that maturity.


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Futures Pricing

Pricing Futures: Considerations


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Pricing Futures: Considerations

Analytical valuation of futures contracts difficult for two reasons:

1. Delivery options provided to the short position.

2. Margining which creates uncertain interim cash flows.

These features will have an impact on futures prices compared to another

wise identical forward contract.

The question is: how much of an effect? Is it quantitatively significant?

Delivery Options


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Delivery Options

Consider delivery options.

Such options make the futures contract

more attractive to the seller (the short position)

less attractive to the buyer (the long position). Thus, other things being equal the futures price must be lower than the

forward price on this account.

How much lower? That is, how economically valuable is the delivery option?

D li O ti


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Delivery Options

The delivery option is provided primarily to guard against squeezes.

However, provision of the delivery option degrades the quality of the hedge.

Intuitively, the more valuable this option, the greater this uncertainty, and the

more the hedge is degraded.

Thus, we would expect that in a successfulfutures contract, the delivery

option does not have much economicvalue.

Empirical studies support this position: One study of the T-Bond futures

contract, for example, found that the option was worth around $430 even

with 6 months left to maturity.

Ma gin Acco nts


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Margin Accounts

What about margin accounts?

These create interim cash flows which earn interest at possibly uncertain

rates.

Thus, the quantitative impact will depend on

how cash flows into the margin account occur (i.e., how futures prices

change)

how interest rates change when futures prices change (i.e., the

correlation between interest rate changes and futures price changes).

Once again, in a successful futures contract, our expectation would be thatthis impact would be quantitatively small.

Margin Accounts


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Margin Accounts

Best "laboratory" for testing the effect: currency contracts, where no delivery

options exist.

One study reported that difference in forward and futures prices were smallerthan the bid-ask spread in the currency market.

Futures Pricing: Summary


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Futures Pricing: Summary

Identifying futures prices analytically is complicated by

delivery options

margin accounts

It is possible to take these factors into account and develop a full pricing

theory for futures.

However, both theory and empirical evidence point to only a small effect

especially for short-dated contracts.

Given this, we treat futures and forward prices in the sequel as if they were

the same.

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