Determination of Forward and Futures Prices

Chapter 5

Consumption vs Investment Assets

Investment Assets: That is held for investment purposes by significant

numbers of investors.

(Examples: stocks, bonds, gold, silver) Consumption Assets: That is held by primarily for consumption.

(Examples: copper, oil, pork)

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

Required to maintain a margin account with the broker

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

Cash flows form short sale and purchase of shares

Purchase of shares April : Purchase 500 shares for $120 -$60,000 May : Receive dividend +$500 July : Sell 500 shares for $100 per share +$50,000 Net profit= -$9,500

-----------------------------------------------------------------------------Short sale of shares

April : Borrow 500 shares and sell them for $120 +$60,000 May : Pay dividend - $500 July : Buy 500 shares for $100 per share -$50,000 Replace borrowed shared to short position Net profit= +$9,500

Assumption and Notation

Assumption: 1.No transaction costs when they trade.

2.The same tax rate on all net trading profits.

3.Borrow money at the same risk-free rate

of as they can lend money.

4.Take advantage of arbitrage opportunities

as they occur.

Assumption and Notation

S0: Price of the asset underlying the forward or futures contract today

F0: Futures or forward price today

T: Time until delivery date

r: Risk-free interest rate for maturity T

﹡NOTATION:

Forward Price For an Investment Asset

Assume : S0 = $40,r = 5%,t = 3 months

(a)If F0 =$43 > S0ert

1.Borrow $40 at risk-free interest rate of 5% per annum.

2.Short a forward contract to sell one share in 3-months.

$40e0.05x3/12 = $40.5

$43 - $40.5 = $2.5

(b)If F0 =$39 < S0ert

1.Short one share, invest the proceeds of the short sale

at 5% per annum for 3 months.

2.Take a long position in a 3-months forward contract.

$40e0.05x3/12 = $40.5

$40.5 - $39 = $1.5

∴We deduce that for there to be no arbitrage the forward

price must be exactly $40.5.

F0 = S0erT

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

What If Short Sale Are Not Possible? (a)If F0 > S0ert

1.Borrow S0 dollars at an interest rate r for T years.

2.Buy 1 ounce of gold.

3.Short a forward contract on 1 ounce of gold.

The investor make a profit of F0 - S0ert.

(b)If F0 < S0ert

1.Sell the gold for S0.

2.Invest the proceeds at interest rate r for time T.

3.Take a long position in a forward contract on

1 ounce of gold.

The investor make a profit of S0ert - F0.

When an Investment Asset Provides a Known Dollar Income

F0 = (S0 – I )erT

where I is the present value of the income during life of forward contract

Known Income

Assume : S0 = $900 I = 40e-0.03x4/12 = $39.6 r =0.04 T = 0.75(9/12) I: ? $40 0 4 9

F0 = (900.00 – 39.6)e0.04x0.75 = $886.60

(a)If F0 = $910 > (S0 - I)ert = $886.60

1.Borrow $900 to buy the bond.

2.Short a forward contract.

→ 900.00 - 39.6 = $860.40

→ 860.40e0.04x0.75 = $886.60

→ 910.00 -886.60 = $23.40

(b)If F0 = $870 < (S0 - I)ert = $886.60

1.Short the bond.

2.Enter into a long forward contract.

→ 900 - 39.6 = $ 860.4

→ 860.40e0.04x0.75 = $886.60

→ 886.60 - 870 = $16.60

∴ The forward price must be $886.60

Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

5.15

When an Investment Asset Provides a Known Yield

F0 = S0 e(r–q )T

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

Known Yield

Assume : S0 = 25,r = 0.1,and T = 0.5,

the yield is 4% per annum with semiannual

compounding.

1+0.04 = (1+q/2)2 q = 3.96%

F0 = 25e(0.10 – 0.0396)x0.5 = $25.77

Valuing a Forward Contract

K is delivery price in a forward contract

F0 is forward price today ƒ :Value of forward 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

The value of a forward contract on an investment asset that provides no income:

ƒ = (F0–K)e-rt

Equation shows that F0 = S0ert

ƒ = (S0ert–K)e-rt

ƒ = S0 – Ke-rt

The value of a long forward contract on an investment asset that provides a known income with present value I:

ƒ = S0 – I – Ke-rt

The value of a long forward contract on an investment asset that provides a known yield at rate q:

ƒ = S0e-qt – Ke-rt

Forward vs Futures Prices

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 Last only a few months are in most circumstances

sufficiently small to be ignored Forward and futures prices are usually assumed

to be the same. When interest rates are uncertain

they are, in theory, slightly different

Futures Prices Of Stock Index 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 average dividend yield on the portfolio represented by the index during life of contract

Futures Prices Of Stock Index

F0 = S0e(r-q)T

q:The dividend yield

Example:

r = 0.05 S0 = 1,300 T = 3/12 (0.25) q = 0.01

F0 = 1,300e(0.05-0.01)x0.25 = $1,313.07

Index Arbitrage

If F0 > S0e(r-q)T

1.Buying the stocks underlying the index at

the spot price

2.Shorting futures contracts

By a corporation holding short-term money

market investment.

Index Arbitrage

If F0 < S0e(r-q)T

1.Shorting or selling the stocks underlying

the index

2.Taking a long position in futures contracts

By a pension fund that owns an indexed

portfolio of stocks

Index Arbitrage

Program trading Occasionally (e.g., on Black Monday)

simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 does not hold

Futures and Forwards on Currencies

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

Two ways of converting 1,000 units of a foreign currency to dollars at time T. Here, S0 is spot exchange rate, F0 is forward exchange rate, and r and rf are the dollar and foreign risk-free rates.

Futures and Forwards on Currencies

1,000 erfT F0 = 1,000 S0 erT

F0 = S0 erT / erfT

The relationship between F0 and S0

F S e r r Tf

0 0 ( )

Futures on Commodities

Income and Storage Costs(a)In the absence of storage costs and income, the forward price of a commodity that is an investment asset is give by:

F0 = S0erT

(b)If U is the present value of all the storage costs, net of income, during the life of a forward contract:

F0 = (S0 + U)erT

(c)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:

F0=S0e(r+u)T

Where u denotes the storage costs per annum as proportion of the spot price net of any yield earned on the asset.

(a) F0 > (S0 + U)erT

1. Borrow an amount S0 + U at the risk-free rate and

use it to purchase one unit of the commodity and

to pay storage costs.

2. Short a forward contract on one unit of the

commodity.

Futures on Consumption Assets

(b) F0 < (S0 + U)erT

1. Sell the commodity, save the storage costs,

and invest the proceeds at the risk-free

interest rate.

2 . Take a long position in a forward contract.

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.

Convenience Yield

* The benefits from holding the physical asset are sometimes referred to as the convenience yield.

If the dollar amount of storage costs is known and has a present value U, that the convenience yield y is defined such that:

F0eyT = ( S0 + U )erT

If the storage costs per unit are a constant proportion, u, of the spot price, then y is defined so that:

F0eyT = S0e(r+u)T or F0 = S0e(r+u-y)T

The Cost of Carry

The cost of carry, c, is the storage cost plus

the interest costs less the income earned

For a non-dividend-paying stock, it is r. For a stock index, it is r - q. For a currency, it is r - rf. For a commodity that provide income at rate q and

require storage costs at rate u, it is r - q + u.

Define the cost of carry as c. For an investment asset , the futures price is

F0 = S0ecT For a consumption asset, The convenience yield

on the consumption asset, y, is defined

so that

F0 = S0 e(c–y )T

The Cost of Carry

Delivery Options

Form equation (F0 = S0 e(c–y )T ) that c > y, the benefits from holding the asset (including convenience yield and net of storage costs) are less than the risk-free rate .

If futures prices are decreasing as time to maturity increase (c < y).It is then usually optimal for the party with the short position to deliver as late as possible, and futures prices should, as a rule, be calculated on this assumption

The Risk in a Futures Position

The cash flow to the speculator are as follow : Today : - F0e-rT

End of futures contract : +ST

The futures prices today : F0

The prices of the asset at time T : ST

The risk-free return on funds invested for time : T The investor's required return : k The expected value : E The PV of this investment : - F0e-rT

+ E(ST)e-kT

Assume net present value = 0 - F0e-rT

+ E(ST)e-kT = 0 F0 = E(ST)e(r-k)T

The Risk in a Futures Position

If the asset has no systematic risk, then k = r F0 = E(ST)

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 )

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.