3.1

Determination of Forward and

Futures Prices

Chapter 3

3.2Determinants of Forward and Futures Prices

• Examine forward prices first and then show how forward and futures prices are related

• Forward prices on investment assets– that provide no income

• e.g. non-dividend paying stock– that provide a known cash income

• e.g. coupon bond or – that provide a known dividend yield

• Forward prices on consumption assets

3.3

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)

3.4

Short Selling (Page 40-42)

• 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

3.5

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

3.6

Some Preliminaries: Measuring Interest Rates

• The compounding frequency used for an interest rate is the unit of measurement

• The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers

3.7Some Preliminaries: Continuous

Compounding(Page 43)

• In the limit as we compound more and more frequently we obtain continuously compounded interest rates

• $100 grows to $100eRT when invested at a continuously compounded rate R for time T

• $100 received at time T discounts to $100e-RT

at time zero when the continuously compounded discount rate is R

3.8

Conversion Formulas(Page 43)

Define

Rc : continuously compounded rate

Rm: same rate with compounding m times per year

R m

R

m

R m e

cm

mR mc

ln

/

1

1

3.9

Notation

S0: Spot price today

F0: Futures or forward price today

T: Time until delivery date

r: Risk-free interest rate for maturity T

3.10

Gold Example (From Slide 1.26)

• For gold

F0 = S0(1 + r )T

(assuming no storage costs)• If r is compounded continuously instead

of annually

F0 = S0erT

3.11

Extension of the Gold Example(Page 46, equation 3.5)

• For any investment asset that provides no income and has no storage costs

F0 = S0erT

3.12When an Investment Asset Provides a Known Dollar Income (page 49, equation 3.6)

F0 = (S0 – I )erT

where I is the present value of the income

3.13

When an Investment Asset Provides a Known Yield

(Page 51, equation 3.7)

F0 = S0 e(r–q )T

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

3.14Valuing a Forward ContractPage 51

• The value of a forward contract at the time when it is first entered into is zero.

• Value of forward contract after the initiation of the contract– Suppose that

K is delivery price in a forward contract & F0 is forward price that would apply to the contract

today– The value of a long forward contract, ƒ, is ƒ = (F0 – K )e–rT

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

3.15

Forward prices:Forwards as a delayed sale

• An investor is considering the sale of a stock three months from today:– Immediate sale: sell the stock today at the current

spot price for the asset, S0 (e.g. S0 = 61.00).• deposit S0 and earn the interest rate r per year with

continuous compounding (e.g. r = 6.71%)– Delayed sale: enter into a forward contract today

to sell the stock in three months for F0.• What is the least that the investor is willing to

receive for the stock in three months?

3.16

Forward prices:Forwards as a delayed purchase

• An investor is considering the purchase of a stock three months from today:– Immediate purchase: buy the stock today at the

current spot price for the asset, S0 (e.g. 61.00)– Delayed purchase: Take the current price of the

stock today, S0, and deposit it and earn the interest rate r per year with continuous compounding (e.g. r = 6.71%)

• What is the most that the investor is willing to pay for the stock in three months?

3.17

Forward prices:Forwards as a delayed sale or purchase• Putting it together: If there are to be no

arbitrage opportunities in the market, what is the forward price, F, for a contract to buy or sell this stock three months from today?

• Suppose the stock pays income to an investor between today and the maturity of the contract?– How should we, if at all, adjust the price?

• Think about the buyer• Think about the seller• Think about whether the asset is a financial asset or

a commodity for consumption

3.18Forward contract on Types of Assets: Each asset has an associated cost of carrying the asset

from the spot to forward market

Financial Investments• Investments that provide no income• Investments that provide a known cash income• Investments that provide a known dividend yield• Commodities for investment with storage costs (e.g.,

gold and silver)

Consumption Investments• Commodities with a known cash storage cost• Commodities with a known continuous storage cost• Convenience yields

3.19

No Arbitrage Pricing• Consider a six-month forward contract on a dividend

paying stock with a price of $34.375 per share. – Dividends of $0.20 per share are expected after three and

six months.• The three-month and six-month risk-free rates of

interest are, respectively, 7.875% and 8.00% per year with continuous compounding.

• The quoted forward price for a contract maturing in six-months is $35.80.

• Does an arbitrage opportunity exist?

3.20

Forward vs Futures Prices• Forward and futures prices are usually assumed

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

• A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price

• A strong negative correlation implies the reverse

3.21

Stock Index• Stock index tracks the change in the value of a

hypothetical portfolio of stocks.– S&P 500 Index is based on a portfolio of 500 stocks (400

industrial, 40 utilities, 20 transportation, and 40 financial companies.)

– Nikkei 225 Stock Average is based on a portfolio of 225 of the largest stocks trading on the Tokyo Stock Exchange

• Stock index futures: futures contract on the stock index– S&P 500 Futures is on 250 times the index– Nikkei 225 Stock Average is on 5 times the index

3.22

Stock Index continued• 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• For the formula to be true it is important that the

index represent an investment asset• In other words, changes in the index must

correspond to changes in the value of a tradable portfolio

• The Nikkei index viewed as a dollar number does not represent an investment asset

3.23

Stock Index Arbitrage• The purchase or sale of a portfolio of stocks that

replicates a stock index and the sale or purchase of a futures contract on the index

• Occurs when the F0 does not equal S0e(r-q)T

– When F0>S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures

– When F0<S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index

• Often implemented through program trading– execution on a stock market of a large number of

simultaneous buy or sell orders.– Triggered by a computer program that detects an arbitrage

opportunity or suggests some other reason for quickly establishing a large portfolio of stocks.

3.24

Index Arbitrage(continued)

• Index arbitrage involves simultaneous trades in futures and many different stocks

• Very often a computer is used to generate the trades

• Occasionally (e.g., on Black Monday) simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 does not hold

3.25

Stock Index Arbitrage

• On November 8, the S&P 500 Index is at $1150.50, the dividend yield is 3% and the risk-free rate is 5%. The December futures contract is priced at $1,160.46 and expires on December 18. T = 40/365 = 0.1096

• What is the theoretical price of the December futures contract?

• What type of an arbitrage can be executed using $20 million.

• On December 18 the S&P 500 Index is at 1,146.41.

3.26

• A foreign currency is analogous to a security providing a dividend yield

• The continuous dividend yield is the foreign risk-free interest rate

• It follows that if rf is the foreign risk-free interest rate

Futures and Forwards on Currencies (Page 57-59)

F S e r r Tf

0 0 ( )

3.27Futures on Consumption Assets

(Page 62)

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.

3.28

Convenience Yields

• Benefits to users of commodity that are not obtained by holders of futures contracts– Ability to keep a production process running– Ability to profit from temporary local shortages

• The convenience yield, y, is defined so that– If storage costs are a known cash amount

F0eyT = (S0+U)erT

– If storage costs are proportional to the commodity price

F0 = S0e(r+u-y)T

3.29

The Cost of Carry (Page 63)

• 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

3.30

Cost of Carry: Summarizes relation between futures and spot price

Underlying Assets Cost of Carry (c)

Non-dividend paying stock or non-coupon paying bond

Stock or asset with known cash income (ie., not proportional to asset price)

Stock index or asset with known dividend yield

Foreign currency

Commodity with storage costs proportional to price

Consumption asset

3.31

Cost of Carry PricingUnderlying Assets F0 = S0e

cT c = cost of carry

Non-dividend paying stock or non-coupon paying bond

F0 = S0erT

Stock or asset with known cash income (ie., not proportional to asset price)

F0 = (S0 – I )erT

Where I = p.v. income

Stock index or asset with known dividend yield

F0 = S0e(r – q)T

Foreign currency F0 = S0e(r – r

f)T

Commodity not f or consumption with storage costs proportional to price

F0 = S0e(r + u)T

Consumption asset F0 < = S0e(r + u)T,

F0 = S0e(c – y)T

where c = r + u

3.32

Futures Prices & Expected Future Spot Prices (Page 64)

• Suppose k is the expected return required by investors on an asset

• We can invest F0e–r T now to get ST back at maturity of the futures contract

• This shows that

F0 = E (ST )e(r–k )T

3.33Futures Prices & Future Spot Prices (continued)

• If the asset has –no systematic risk, then

k = r and F0 is an unbiased estimate of ST

–positive systematic risk, then

k > r and F0 < E (ST )–negative systematic risk, then

k < r and F0 > E (ST )

3.34

Convergence of Futures Price to Spot Priceand Backwardation and Contango

Time Time

(b)

FuturesPrice

FuturesPrice

Spot Price

Spot Price

(a)