Chuong 2 Forward and Futures 2013 S

Chapter 2

FORWARD AND FUTURES CONTRACTS

(12 hours)

Outline

I. Mechanics of futures markets

II. Hedging strategies using futures

III. Determination of forward and futures prices

I. Mechanics of Futures market

• Specification of a futures market• Margins• Closing out positions• Delivery• Some terminology • Patterns of futures price• Forward and futures contracts

Specification of a futures contract

• Asset• Contract size• The time and place of delivery

Margins

• A margin is cash or marketable securities deposited by an investor with his or her broker

• The balance in the margin account is adjusted to reflect daily settlement

• Margins minimize the possibility of a loss through a default on a contract

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Example of a Futures Trade

• An investor takes a long position in 2 December gold futures contracts on June 5– contract size is 100 oz.– futures price is US$900– margin requirement is US$2,000/contract (US$4,000

in total)– maintenance margin is US$1,500/contract (US$3,000

in total)

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Fundamentals of Futures and Options Markets, 7th Ed, Ch 2, Copyright © John C.

Hull 2010

A Possible OutcomeTable 2.1, Page 27

Daily Cumulative Margin

Futures Gain Gain Account Margin

Price (Loss) (Loss) Balance Call

Day (US$) (US$) (US$) (US$) (US$)

900.00 4,000

5-Jun 897.00 (600) (600) 3,400 0. . . . . .. . . . . .. . . . . .

13-Jun 893.30 (420) (1,340) 2,660 1,340 . . . . . .. . . . .. . . . . .

19-Jun 887.00 (1,140) (2,600) 2,740 1,260 . . . . . .. . . . . .. . . . . .

26-Jun 892.30 260 (1,540) 5,060 0

+

= 4,000+

= 4,000

7

Other Key Points About Futures

• They are settled daily• Closing out a futures position involves

entering into an offsetting trade• Most contracts are closed out before

maturity

8


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Delivery• The vast majority of futures contracts do not lead to delivery.

Most of traders choose to close out their positions prior to maturity.

• If a futures contract is not closed out before maturity, it is usually settled by delivering the assets underlying the contract. When there are alternatives about what is delivered, where it is delivered, and when it is delivered, the party with the short position chooses.

• A few contracts (for example, those on stock indices and Eurodollars) are settled in cash

• When there is cash settlement contracts are traded until a predetermined time. All are then declared to be closed out.

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Some Terminology

• Open interest: the total number of contracts outstanding. This equals to number of long positions or number of short positions

• Settlement price: the price just before the final bell each day. This is used for the daily settlement process

• Volume of trading: the number of trades in 1 day

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Questions

• When a new trade is completed what are the possible effects on the open interest?

• Can the volume of trading in a day be greater than the open interest?

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Convergence of Futures to Spot (Figure 2.1, page 25)

Time Time

(a) (b)

FuturesPrice

FuturesPrice

Spot Price

Spot Price

12

Patterns of futures price

• Normal market: The settlement futures prices increase with the maturity of the contract.

• Inverted market: the settlement futures prices decrease with the maturity of the contract.

Futures for Crude Oil on Aug 4, 2009


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Futures for Soybeans on Aug 4, 2009


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Forward Contracts

• A forward contract is an OTC agreement to buy or sell an asset at a certain time in the future for a certain price

• There is no daily settlement (but collateral may have to be posted). At the end of the life of the contract one party buys the asset for the agreed price from the other party

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Profit from a Long Forward or Futures Position

Profit

Price of Underlying at Maturity

17

Profit from a Short Forward or Futures Position

Profit

Price of Underlying at Maturity

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Forward Contracts vs Futures Contracts (Table 2.3, page 40)

Forward Futures

Private contract between two parties Traded on an exchange

Not standardized Standardized

Usually one specified delivery date Range of delivery dates

Settled at end of contract Settled daily

Delivery or final settlement usual Usually closed out prior to maturity

Some credit risk Virtually no credit risk

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II. Hedging strategies using futures1. Basic principles

1.1. Short hedge

1.2.Long hedge

1.3. Basis risk

1.4. Choice of contract

2. Cross hedging

3. Stock index futures

3.1. Hedging an equity portfolio

3.2. Changing the beta of a portfolio

Long & Short Hedges

• A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price

• A short futures hedge is appropriate when you know you will sell an asset in the future & want to lock in the price

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Short Hedges- Example (a)• It is May 15 today and an oil producer has just

negotiated a contract to sell 1 million barrels of crude oil. It has been agreed that the price that will apply in the contract is the market price on August 15. How does the firm hedge its exposure?

• Suppose that on May 15, the spot price is $60 per barrel and the crude oil futures price on the NYMEX for August delivery is $59 per barrel. Each futures contract on NYMEX is for the delivery of 1,000 barrels.

• Short hedge: the company can hedge its exposure by shorting 1,000 futures contracts for August delivery and close its position on August 15.

Short Hedges- Example (b)• On August 15 (1) Suppose the spot price is $55 per barrel: The firm realizes $55 million for the oil under its sales

contract. August is the delivery month for the futures contract The

futures price on August 15 should be very close to the spot price of $55 on that date. The firm gains approximately 59 - 55= $4 per barrel or $4 million in total from the short futures position.

Total amount realized from both the futures position and the sales contract is approximately $59 per barrel or $59 million in total.

Short Hedges- Example (c)• On August 15(2) Suppose the spot price is $65 per barrel: The firm realizes $65 million for the oil under its sales

contract. August is the delivery month for the futures contract

The futures price on August 15 should be very close to the spot price of $65 on that date. The firm losses approximately 65 - 59= $6 per barrel or $6 million in total from the short futures position.

Total amount realized from both the futures position and the sales contract is approximately $59 per barrel or $59 million in total.

Long Hedges- Example (a)• It is now January 15. A copper fabricator

knows it will require 100,000 pounds of copper on May 15 to meet a certain contract. How does the firm hedge its exposure?

• Suppose that the spot price of copper is 340 cents per pound, and the futures price for May delivery is 320 cents per pound.

• Long hedge: the firm can hedge its exposure by taking a long position in four futures contract on COMEX division of NYMEX and closing its position on May 15.

Long Hedges- Example (b)• On May 15:(1) Suppose the spot price is 325 cents per pound. The firm pays: 100,000 x$3.25= $325,000 for the

copper in the spot market. May is the delivery month for the futures contract

The futures price on May 15 should be very close to the spot price of 325 cents on that date. The firm gains approximately 325 - 320= 5 cents per pound or $5000 on the futures contract.

It net cost is approximately $320,000 or 320 cents per pound.

Long Hedges- Example (c)• On May 15:(1) Suppose the spot price is 305 cents per pound. The firm pays: 100,000 x$3.05= $305,000 for the

copper in the spot market. May is the delivery month for the futures contract

The futures price on May 15 should be very close to the spot price of 305 cents on that date. The firm losses approximately 320- 305= 15 cents per pound or $15000 on the futures contract.

It net cost is approximately $320,000 or 320 cents per pound.

• Basis risk:

In practice, hedging is often not quite as straightforward, because:

The asset whose price is to be hedged may not exactly the same as the asset underlying the futures contract.

The hedger may be uncertain as to the exact date when the asset will be bought or sold.

The hedge may require the futures contract to be closed out before its delivery month.

Basis risk often exists.

Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright © John C.

Hull 2010

Convergence of Futures to Spot(Hedge initiated at time t1 and closed out at time t2)

Time

Spot Price

FuturesPrice

t1 t2

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Basis Risk• Basis is the difference between spot &

futures: Basis= Spot price of asset to be hedged –

Futures price of contract used• Basis risk arises because of the uncertainty

about the basis when the hedge is closed out

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Long Hedge • Suppose that

F1 : Initial Futures Price (time t1)F2 : Final Futures Price (time t2)S2 : Final Spot Price (time t2)Basis=b2 = S2-F2

• You hedge the future purchase of an asset by entering into a long futures contract

• Cost of Asset=S2 – (F2 – F1) = F1 + Basis• If basis increases unexpectedly , the hedger’s

position worsen; if basis decreases unexpectedly, the hedger’s position improves.

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Short Hedge• Suppose that

F1 : Initial Futures Price (time t1)

F2 : Final Futures Price (time t2)

S2 : Final Spot Price (time t2)

Basis= b2= S2-F2• You hedge the future sale of an asset by

entering into a short futures contract• Price Realized=S2+ (F1 – F2) = F1 + Basis• If basis increases, the hedger’s position

improves unexpectedly; if basis decreases unexpectedly, the hedger’s position worsens. 32

If the asset underlying the futures contract is not the same as the asset whose price is being hedged:

• S2* : spot price of the underlying asset at time t2 :• F1-F2 +S2 = F1 + (S2*-F2)+(S2-S2*)Basis composes of two components:

S2*-F2 : the difference between spot & futures of the underlying asset.

S2-S2* : the difference between the spot prices of the two assets.

•

Problems1.On March 1 the spot price of a commodity is $20 and the July

futures price is $19. On June 1 the spot price is $24 and the July futures price is $23.50. A company entered into a futures contracts on March 1 to hedge the purchase of the commodity on June 1. It closed out its position on June 1. What is the effective price paid by the company for the commodity?

2. On March 1 the price of a commodity is $300 and the December futures price is $315. On November 1 the price is $280 and the December futures price is $281. A producer entered into a December futures contracts on March 1 to hedge the sale of the commodity on November 1. It closed out its position on November 1. What is the effective price received by the producer?

Choice of Contract (a)

• The choice of the asset underlying the futures contract.

• The choice of the delivery month

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Choice of Contract (b)

• Choose a delivery month that is as close as possible to, but later than, the end of the life of the hedge

• When there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price.

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Choice of Contract- Example (a)• It is March 1. A US company expects to receive 50

million Japanese yen at the end of July. Yen future contracts on the CME have delivery months of March, June, September and December. One contract is for the delivery of 12.5 million yen.

• The company shorts four September yen futures contract on March 1. When the yen are received at the end of July, the company closes out its position.

• The futures price on March 1 in cents per yen is 0.7800 and the spot and futures prices when the contract is closed out are 0.72000 and 0.7250, respectively.

Choice of Contract- Example (b)• The gain on the futures contract: 0.7800-0.7250 =

0.0550 cents per yen. • The basis: 0.72000-0.7250= -0.0050. • The effective price obtained in cents per year:The final spot price + the gain 0.7200 + 0.0550 = 0.7750 Or the initial future price + the final basis 0.7800 + (-0.0050) = 0.7750

Cross hedging:

• When the asset underlying the futures contract is

not the same as the asset to be hedged.

• Ex: An airline is concerned about the future price

of jet fuel. Because there is no futures contract on

jet fuel, it might choose to use heating oil futures

contracts to hedge its exposure.

• Hedge ratio: is the ratio of the size of the

position taken in futures contracts to the size of

the exposure.

• When the asset underlying the futures contract

is the same as the asset being hedged, the

hedge ratio equals 1.

• The hedger should choose a value for the hedge

ratio that minimizes the variance of the value of

the hedged position.


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Optimal Hedge Ratio

Proportion of the exposure that should optimally be hedged is

where sS is the standard deviation of DS, the change in the spot price during the hedging period, sF is the standard deviation of DF, the change in the futures price during the hedging periodr is the coefficient of correlation between DS and DF.

(See appendix, page 71)

F

Sh*

41

• How to calculate h*?

- Using data on the changes in historical spot and futures prices.

+ Choose a number of equal non-overlapping time intervals. Ideally, the length of each interval is the same as the length of the time interval for which the hedge is in effect.

+ Observe the changes in spot price of the asset being hedged and the changes in future price of the underlying asset. Compute the standard deviations of these changes.

Optimal number of contracts- QA: Size of position being hedged (units)- QF: Size of one futures contract (units)

- N* : Optimal number of futures contracts for hedging.

F

A

Q

QhN **

• Ex: An airline expects to purchase 2 million gallons of jet fuel in 1 month and decides to use heating oil futures for hedging. The size of one heating oil futures contract is 42,000 gallons Compute the optimal number of futures contracts for hedging.

• Data on the changes in historical prices:Month

(i)Changes in heating oil futures price per gallon

(xi)

Changes in jet fuel price per gallon

(yi)

123456789

101112121415

0.0210.035-0.0460.0010.044-0.029-0.026-0.0290.048-0.006-0.036-0.0110.019-0.0270.029

0.0290.020-0.0440.0080.026-0.019-0.010-0.0070.0430.011-0.036-0.0180.009-0.0320.023

Method 1: Calculate mean, standard deviation, correlation of the changes in the spot and futures prices.

h= 0.928x(0.0263/0.0313) = 0.78

Method 2: h* is the slope of the best-fit line when deltaS is regressed against deltaF.

deltaS = 0.78 deltaF

0313.0F 0263.0S

928.0/),( SFSFCov

• The optimal number of futures contracts for hedging:

0.78 x 2,000,000/42,000 = 37.14

Problems3. Suppose that the standard deviation of quarterly changes in

the prices of a commodity $0.65, the standard deviation of quarterly changes in a futures price on the commodity is $0.81, and the coefficient of correlation between the two changes is 0.8. What is the optimal hedge ratio for a 3-month contract? What does it mean?

4. The standard deviation of monthly changes in the spot price of live cattle is (in cents per pound) 1.2. The standard deviation of monthly changes in the futures price of live cattle for the closest contract is 1.4. The correlation between the futures price changes and the spot price changes is 0.7. A beef producer is committed to purchasing 200,000 pounds of live cattle next month. The producer wants to use the live cattle futures contracts to hedge its risk. Each contract of/ for the delivery of 40,000 pounds of cattle. What strategy should the beef producer follow?

• Stock index futures- A stock index tracks changes in the value of a

hypothetical portfolio of stocks. - The weight assigned to the stocks are

proportional to their market prices or their market capitalizations.

- Return on the stock index is usually used as a proxy of the market return. The stock index is given a beta of 1.

- Some exchanges provide futures contracts for underlying assets of stock indices. For example: Dow Jones Industrial Average, S&P500, Nasdaq 100, Russell 100, U.S. Dollar Index futures contracts.


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Hedging Using Index Futures

To hedge the risk in a portfolio the number of contracts that should be shorted is

where VA is the current value of the portfolio, b is its beta, and VF is the current value of one futures (=futures price times contract size)

F

A

V

VN b*

50

• Example (a) : Suppose that a futures contract with 4 months to maturity is used to hedge the value of a portfolio over the next 3 months:

- Value of portfolio= $5,050,000- Beta of portfolio= 1.5 - Value of S&P 500 index= 1000- S&P 500 futures price= 1,010 USD

- Risk-free interest rate =4% per annum- Dividend yield on index =1% per annum

- One contract is on $250 times the index - What position in futures contracts on the S&P 500

is necessary to hedge the portfolio?

Example (b)

- The current value of one futures F = 250 x 1010 = 252,500

- The current value of the portfolio: 5,050,000 USD

- To hedge the risk in a portfolio the number of contracts that should be shorted :

1.5 x (5,050,000/ 252,500) = 30

Example (c)

• Suppose that the index turns out to be 900 in 3 months and the futures price is 902. Calculate the value of the hedged portfolio in 3 months?

• Example (d)• The index futures price decreases to 902 in 3 months. The gain from the short futures position:

30 x(1010-902) x 250 = 810,000The index turns out to be 900 in 3 months The loss on the

index is 10%. The index pays a dividend of 1% per annum, or 0.25% per three months. When dividends are taken into account, an investor in the index would earn -9.75%.

The portfolio has beta=1.5, risk-free rate= 1% per 3 monthsThe expected return on the portfolio during the 3 months: 1

+ 1.5( -9.75-1 ) = -15.125%Expected value of the portfolio: 5,050,000 x (1-15,125%) =

$4,286,187The expected value of the hedger’s position: 810,000 +

4,286,187= $5,096,187

Problems

5. On July 1, an investor holds 50,000 shares of a certain stock. The market price is $30 per share. The investor is interested in hedging against movements in the market over the next month and decides to use the September Mini S&P 500 futures contract. The index futures price is currently 1,500 and one contract is for delivery of $50 times the index. The beta of the stock is 1.3. What strategy should the investor follow?

6. A fund manager has a portfolio worth $50 million with a beta of 0.87. The manager is concerned about the performance of the market over the next 2 months and plans to use 3-month futures contracts on the S&P 500 to hedge the risk. The current level of the index is 1,250, one contract is on 250 times the index, the risk-free rate is 6% per annum, and the dividend yield on the index is 3% per annum. The current 3-month futures prices is 1259.

• What position should the fund manager take to hedge all exposure to the market over the next 2 months?

• Suppose that in 2 months, the index decreases to 1000, and the index futures price is 1002.5. Calculate the expected value of the company after hedging the risk.

Changing the beta of a portfolio• Sometimes futures contracts are used to

change the beta of a portfolio to some value other than zero.

- To reduce the beta, the number of futures contract should be shorted:

- To increase the beta, the number of futures contracts should be taken in a long position:

F

PN )( ** bb *bb

F

PN )( ** bb *bb

• Example- Value of portfolio= $5,050,000- Beta of portfolio= 1.5 - Value of S&P 500 index= 1000- S&P 500 futures price= 1,010 USD- One contract is on $250 times the index

What position is necessary to reduce the beta of the portfolio to 0.75?

What position is necessary to increase the beta of the portfolio to 2.0?

Problems7. A company has a $20 million portfolio with a beta of

1.2. It would like to use futures contracts on the S&P 500 to hedge its risk. The index futures price is currently standing at 1080, and each contract is for delivery of $250 times the index. What is the hedge that minimizes risk? What should the company do if it wants to reduce the beta of the portfolio to 0.6?

Problems

8. It is July 16. A company has a portfolio of stocks worth $100 million. The beta of the portfolio is 1.2. The company would like to use the CME December futures contract on the S&P 500 to change the beta of the portfolio to 0.5, during the period July 16 to November 16. The index futures price is currently 1,000, and each contract is on $250 times the index.

- What position should the company take? - Suppose that the company changes its mind and

decides to increase the beta of the portfolio from 1.2 to 1.5. What position in futures contracts should it take?

IV. Determination of Forward and Futures Prices

1. Forward price

1.1. Assumption and notation

1.2. Forward price for an investment asset

1.3. Valuing a forward contract

1.4. Are forward prices and futures prices equal?

2. Some specific forward/futures contracts

2.1. Futures on stock indices

2.2. Forward and futures contract on currencies

2.3. Futures on commodities

• Assumptions and notation- The market participants are subject to no

transaction costs when they trade.- The market participants are subject to the

same tax rate on all net trading profits. - The market participants can borrow money

at the same risk-free rate of interest as they can lend money.

- The market participants take advantage of arbitrage opportunities as they occur.

Notation

S0: Spot price today

F0: Futures or forward price today

T: Time until delivery date

r: Risk-free interest rate for maturity T

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Consumption vs Investment Assets

• Investment assets are assets held by significant numbers of people purely for investment purposes (Examples: gold, silver)

• Consumption assets are assets held primarily for consumption (Examples: copper, oil)

64

Short Selling

• Short selling involves selling securities you do not own

• Your broker borrows the securities from another client and sells them in the market in the usual way

65

Short Selling(continued)

• At some stage you must buy the securities back so they can be replaced in the account of the client

• You must pay dividends and other benefits the owner of the securities receives

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Arbitrage• The simultaneous buying and selling of

securities, currency, or commodities in different markets or in derivative forms in order to take advantage of differing prices for the same asset. (Oxford Dictionary)

• An arbitrage is a transaction that involves no negative cash flow at any probabilistic or temporal state and a positive cash flow in at least one state; in simple terms, it is the possibility of a risk-free profit at zero cost.

Forward price for an investment asset• Example (a): Consider a long forward

contract to purchase a non-dividend-paying stock in 3 months. The current stock price is $40 and the 3-month risk-free interest rate is 5% per annum. The forward price is relatively high at $43. What arbitrage opportunities does this create?

• Example (b):

An arbitrageur can:

Action now:

Borrow $40 at 5% for 3 months

Buy one unit of asset.

Enter into forward contract to sell in 3 months for $43.

Action in 3 months:

Sell asset for $43

Use $40.50 to repay loan with interest

Profit realized: $2.50

• Example (c)

Suppose that the forward price is $39.

What arbitrage opportunities does this create?

• Example (d):

An arbitrageur can:

Action now:Short 1 unit of asset to realize $40Invest $40 at 5% for three monthsEnter into a forward contract to buy asset in 3

months for $39.

Action in 3 months:

Buy asset for $39Close short positionReceive $40.50 from investment


A generalization • Forward price:

This equation relates the forward price and the spot price for any investment asset that provides no income and has no storage costs

rTeSF 00

A generalization (b)• Forward price:

• If arbitrageurs can borrow money, buy the asset and short forward contracts on the asset.

• If arbitrageurs can short the asset, invest in the risk-free asset and enter into long forward contracts on the asset.

rTeSF 00

rTeSF 00

rTeSF 00

Problems9. Suppose that you enter into a 6-month

forward contract on a non-dividend-paying stock when the stock price is $30 and the risk-free interest rate (with continuous compounding) is 12% per annum. What is the forward price?

When an Investment Asset Provides a Known Dollar Income

Example: Consider along forward contract to purchase a coupon-bearing bond whose current price is $900 . The forward contract matures in 9 months. A coupon payment of $40 is expected after 4 months. The 4-month and 9-month risk free interest rates (continuously compounded) are 3% and 4%, respectively. The forward price is relatively high at $910. What arbitrage opportunities does this create?

• An arbitrageur can:

Action now- Borrow $900 to buy the bond and short a

forward contract on the bond.- Of the $900, $39.60 is borrowed at 3% per

annum for 4 months so that it can be repaid with the coupon payment. The remaining $860.40 is borrowed at 4% per annum for 9 months.

• Action in 4 months:Receive a coupon payment of $40Use $40 to repay the first loan with

interest• Action in 9 months

Sell the bond for $910Use $886.40 to repay the second loan

with interest


• Suppose that the forward price is relatively low at $870. An arbitrageur can:

Short the bond.

Enter into a forward contract to buy the bond in 9 months for $870.

Known Income- A generalization

Where I is the present value of the income during life of forward contract.

• If buy the asset and short the forward contract.

• If short the asset and take a long position in the forward contract.

rTeISF )( 00

rTeISF )( 00

rTeISF )( 00

• Example: Consider a 10-month forward contract on a stock when the stock price is $50. We assume that the risk-free rate of interest (continuously compounded) is 8% per annum for all maturities. We also assume that dividends of $0.75 per share are expected after 3 months, 6 months, and 9 months. What is the forward price?

Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008 81

When an Investment Asset Provides a Known Yield

(Page 107, equation 5.3)

F0 = S0 e(r–q )T

where q is the average yield during the life of the contract (expressed with continuous compounding)

Method 1: Action now, at time 0:

• Borrow

• Buy N units of the assets and invest the income

from the asset in the asset

• Enter into a forward contract to sell the asset for

F0 per unit at time T.

0NS

Action at time T• Repay the loan: • The income from the asset causes our

holding in the asset to grow at a continuously compounded rate q. By time T, our holding has grown to units of asset.

• Sell the asset for F0:

Net cash flow at time T:

rTeNS0

qTNe

qTeNF0

qTTqr NeeSF )( )(00

At time T- If there are no arbitrate opportunities then

the forward price must be

Or

Where q is the average yield during the life of the contract (expressed with continuous compounding)

0)(00 TqreSF

TqreSF )(00

• If an arbitrageur can

buy the asset and short the forward

contract.

• If an arbitrageur can

short the asset and take a long position in

the forward contract.

TqreSF )(00

TqreSF )(00

Method 2: Action now, at time 0:

• Borrow

• Buy units of the assets and invest the

income from the asset in the asset

• Enter into a forward contract to sell the asset for

F0 per unit at time T.

qTeNS 0

qTNe

Action at time T• Repay the loan: • The income from the asset causes our

holding in the asset to grow at a continuously compounded rate q. By time T, our holding has grown to units of asset.

• Sell the asset for F0:

Net cash flow at time T:

TqreNS )(0

N

0NF

NeSF Tqr )( )(00

At time T- If there are no arbitrate opportunities then

the forward price must be:

Or

Where q is the average yield during the life of the contract (expressed with continuous compounding)

0)(00 TqreSF

TqreSF )(00

Ex: Consider a 6-month forward contract on an asset that is expected to provide income equal to 2% of the asset price once during a 6-month period. The risk-free rate (continuously compounding) is 10% per annum. The asset price is $25 and the forward contract is $30. What should an arbitrageur do?

S0=25, r=0.10, T=0.5, simple yield = 2% continuous yield: q= 2 ln(1+ 0.02) =0.0396

Problems10. Suppose that you enter into a 6-month forward

contract on a non-dividend-paying stock when the stock price is $30 and the risk-free interest rate (with continuous compounding) is 12% per annum. What is the forward price?

11. A stock is expected to pay a dividend of $1 per share in 3 months. The stock price is $50, and the risk-free rate of interest is 10% per annum (with continuous compounding). Consider a long forward contract to purchase the stock. The contract matures in 6 months. Are there any arbitrage opportunities if : (a) the forward price is $49?, (b) the forward price is $53?

12. A stock index currently stands at $350. The risk-free interest rate is 12% per annum (with continuous compounding) and the dividend yield on the index is 4% per annum (with continuous compounding). If the forward price for a 4-month contract is $360. What should an arbitrageur do?

Valuing forward contracts• The value of a forward contract at the time

it is first entered into is zero. • At a later stage, it may prove to have a

positive or negative value. It is important for banks and other financial institutions to value the contract each day.

Valuing forward contracts• Suppose that K is delivery price in a forward contract & F0 is forward price that would apply to the contract

today. ƒ is the value of a long forward contract• At the beginning of the contract life: K = F0 f = 0. • As time passes, K stays the same but the forward price

changes and the value of the long position is : ƒ = (F0 – K )e–rT

• Similarly, the value of a short forward contract is (K – F0 )e–rT

93

• Ex: A long forward contract on a non-dividend-paying stock was entered into some time ago. It currently has 6 months to maturity. The risk-free rate of interest (with continuous compounding) is 10% per annum, the stock price is $25, the delivery price is $24. Calculate the current value of the forward contract?

• S0=25, K=24, r=0.1, T = 0.5• The 6-month forward price:

• The current value of the forward contract:

•

28.2600 rTeSF

17.2)2428.26( 5.01.0 ef

rTKeSf 0

Valuing forward contracts• When the underlying asset provides no

income:

• When the underlying asset provides a known dollar income:

• When the underlying asset provides a known yield:

rTKeSf 0

rTKeISf 0

rTqT KeeSf 0

Forward vs Futures Values • When a futures price changes, the gain or

loss on a futures contract is calculated as the change in the futures price multiplied by the size of the position. This gain/loss is realized almost immediately because of the way futures contracts are settled daily.

• When a forward price changes, the gain or loss is the present value of the change in the forward price multiplied by the size of the position.

Are forward prices and futures prices equal? • Appendix p.126: when interest rates are

constant, forward and futures prices are equal.• Forward and futures prices are usually assumed

to be the same. When interest rates are uncertain they are, in theory, slightly different.

Ex: Suppose a forward contract maturing at time T. The current forward price is G0.

A futures contract on the same underlying asset matures at time T. The current futures price is F0.

rTeSGF 000

• Futures on stock indices• Stock indices can be viewed as an investment

asset paying a dividend yield

• The futures price and spot price relationship is therefore

F0 = S0 e(r–q )T where q is the dividend yield on the portfolio

represented by the index during life of contract

• Ex: Consider a 3-month futures contract on the S&P 500. Suppose that stocks underlying the index provide a dividend yield of 1% per annum. The current value of the index is 1,300. The continuously compounded risk-free interest rate if 5% per annum. Calculate the futures price.

• S0 = 1300, r = 0.05, q = 0.01, T = 0.25

Index Arbitrage• Index arbitrage involves simultaneous trades in

futures and many different stocks • Very often a computer is used to generate the

trades

• If F0 > S0e(r-q)T, profits can be made by buying stocks underlying the index at the spot price and shorting futures contract

• If F0 < S0e(r-q)T profits can be made by shorting the stocks underlying the index and taking a long position in futures contracts..

• A foreign currency is analogous to a security providing a dividend yield. The continuous dividend yield is the foreign risk-free interest rate.

• S0 = the current spot price in dollars of one unit of foreign currency.

• F0 = the forward or futures price in dollars of one unit of foreign currency.

• r = the dollar risk-free rate.

• rf = the foreign risk-free rate.

• The relationship between spot and futures/forward prices:

Futures and Forwards on Currencies

F S e r r Tf

0 0 ( )

102

Foreign Exchange Quotes

• Futures exchange rates are quoted as the number of USD per unit of the foreign currency

• Forward exchange rates are quoted in the same way as spot exchange rates. This means that GBP, EUR, AUD, and NZD are USD per unit of foreign currency. Other currencies (e.g., CAD and JPY) are quoted as units of the foreign currency per USD.

103


Hull 2010

Why the Relation Must Be True Figure 5.1, page 113

1000 units of foreign currency

at time zero

units of foreign currency at time T

Tr fe1000

dollars at time T

Tr feF01000

1000S0 dollars at time zero

dollars at time T

rTeS01000

1000 units of foreign currency

at time zero

units of foreign currency at time T

Tr fe1000

dollars at time T

Tr feF01000

1000S0 dollars at time zero

dollars at time T

rTeS01000

104

Ex: Suppose that the 2-year interest rate in Australia and the US are 5% and 7%, respectively. The spot exchange rate between AUD and USD is 0.6200 USD per AUD. Calculate the 2-year forward exchange rate. Suppose that the 2 year forward exchange rate is 0.6300. What should an arbitrageur do?

• Futures on commodities- Consider futures on commodities that are

investments assets (gold, silver). - Gold and silver provide income to the holder

and also have storage costs. - U= present value of all the storage costs, net of

income, during the life of a forward contract. - Forward/ Futures price:

rTeUSF )( 00

• Ex: Consider a 1-year futures contract on an investment asset that provides no income. It costs $2 per unit to store the asset, with the payment being made at the end of the year. The spot price is $450 per unit and the risk-free rate is 7% per annum for all maturities. Calculate the futures price.

• If the storage costs net of income incurred at any time are proportional to the price of the commodity, they can be treated as negative yield.

• If the above equations do not hold, what should an arbitrageur do?

TureSF )(00

Futures on Consumption Assets

F0 S0 e(r+u )T

where u is the storage cost per unit time as a percent of the asset value.

Alternatively,

F0 (S0+U )erT where U is the present value of the storage

costs.

109

• Convenience yield• The users of a consumption commodity

may feel that ownership of the physical commodity provides benefits that are not obtained by holders of futures contracts. The benefits from holding the physical asset are referred to as the convenience yield provided by the commodity.

• Or

rTyT eUSeF )( 00 TuryT eSeF )(00

TyureSF )(00

The Cost of Carry

• The cost of carry, c, is the storage cost plus the interest costs less the income earned

• For an investment asset F0 = S0ecT

• For a consumption asset F0 S0ecT

• The convenience yield on the consumption asset, y, is defined so that F0 = S0 e(c–y )T

111

• Normal Backwardation and Contango• Normal Backwardation: when the futures

price is below the expected future spot price.

• Contango: when the futures price is above the expected future spot price.

• Normal Backwardation và Contango

Value of a forward contract

rTSeF

Value of a forward Forward price

No income

Known incomeI =PV(Income)

F=(S – I)erT

Known yield q f =S e-qT – K e-rT F = S e(r-q)T

Commodities f=Se(u-y)T- Ke-rT F=Se(r+u-y)T

rTKeSf rTKeISf )(

Problems13. A stock index currently stands at 350. The risk-

free interest rate is 12% per annum (with continuous compounding). The dividend yield on the index is 4% per annum. What should the futures price for a 4-month contract be?

14. The risk-free rate of interest is 7% per annum (with continuous compounding), and the dividend yield on a stock index is 3.2% per annum. The current value of the index is 150. What is the 6-month futures price?

15. A 1-year long forward contract on a non-dividend-paying stock is entered into when the stock price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding.

- What are the forward price and the initial value of the forward contract?

- Six months later, the price of the stock is $45 and the risk-free interest rate is still 10%. What are the forward price and the value of the forward contract?

16. A stock is expected to pay a dividend of $1 per share in 2 months and in 5 months. The stock price is $50 and the risk-free rate of interest is 8% per annum with continuous compounding for all maturities. An investor has taken a short position in a 6-month forward contract on the stock.

- What are the forward price and the initial value of the forward contract?

- Three months later, the price of the stock is $48 and the risk-free rate of interest is still 8% per annum. What are the forward price and the value of the short position in the forward contract?

17. The risk-free interest is 7% per annum with continuous compounding. The dividend yield on a stock index is 4% per annum. The current value of the index is $400. The futures price for a contract deliverable in 4 months is 405. What arbitrage opportunities does this

18. The 2-month interest rate in Switzerland and the US are, respectively, 2% and 5% per annum with continuous compounding. The spot price of the Swiss franc (CHF) is $0.8000. The futures price for a contract deliverable in 2 months is $0.8100. What arbitrage opportunities does this create?

19. The spot price of silver is $9 per ounce. The storage costs are $0.24 per ounce per year payable quarterly in advance. The interest rates are 10% per annum for all maturities. Calculate the futures price of silver for delivery in 9 months.

Chuong 2 Forward and Futures 2013 S

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