note5_determiningforwardprices

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Winter 2014 Determination of Forward and Futures Prices Prof. J. Page

referred to simply as “shorting”, involves selling an asset that is not owned. Short selling

is possible for some, but not all, investment assets.

For example, suppose a trader instructs her broker to short 500 shares of a certain

stock. The broker borrows the shares from another account holder who owns them, and

sells the shares in the market in the usual way. The proceeds from the sale are depositedin the short seller’s account. At some point in the futures, the short seller will close out

her position by repurchasing the 500 shares in the market, which the broker will use to

replace the original borrowed shares. In between the sale and repurchase of the shares,

the short seller is required to pay the amount of any dividends that would have been paid

out by the borrowed shares (or more generally, any income generated by the asset being

shorted). The short seller profits if the share price declines (by more than any dividends

that the shares pay) between the sale and repurchase of the shares. If the share price

increases, the short seller loses money.

Similar to the margin accounts required for futures positions, short sellers must main-

tain a margin account with the broker, consisting of cash or marketable securities to guar-

antee that the trader can cover the short position. As with a futures position, the short

seller may be required to post additional margin if the price of the asset increases beyond

a certain point. If the additional margin is not posted, the short position will be closed

out. In many cases the short seller must also pay a fee for borrowing shares or other

securities.

1.3. Assumptions and Notation

In deriving formulas for forward and futures prices, we will make the following simplifying

assumptions:

1. Market participants face no transaction costs when they trade

2. Market participants are subject to the same tax rate on all net trading profits

3. Market participants can borrow and lend money at the same risk-free rate of interest

4. Market participants can take advantage of arbitrage opportunities as they occur

Note that these assumptions do not need to hold for all market participants. Rather

they only need to be true (or approximately true) for a few key participants such as large

derivatives traders. It is the trading activity of these key market participants and their

eagerness to identify and take advantage of arbitrage opportunities as they occur that

determines the relationship between forward and spot prices.

We will use the following notation throughout:

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T : Time (in years) until the delivery date in the forward or futures contract

S0: Current spot price of the asset underlying the forward or futures contract

F 0: Forward or futures price today

r : Zero coupon risk-free rate for maturity of T years, expressed as an annualized con-

tinuously compounded rate. This is the rate at which money can be borrowed or lent

without default risk. Participants in derivatives markets typically use LIBOR rather

than Treasury rates, because it more closely represents the rates at which financial

institutions and other market participants can actually borrow funds. Since the credit

crisis in 2007, however, many dealers have begun to consider alternatives to LIBOR

as a proxy for the risk-free rate.

Derivatives dealers argue that the interest rates implied by Treasury bills and

Treasury bonds are artificially low because:

1. Treasury bills and Treasury bonds must be purchased by financial institu-

tions to fulfill a variety of regulatory requirements. This increases demand

for these Treasury instruments, driving the price up and the yield down.

2. The amount of capital a bank is required to hold to support an investment

in Treasury bills and bonds is substantially smaller than the capital requiredto support a similar investment in other instruments with very low risk.

3. In the U.S., Treasury instruments are given favorable tax treatment com-

pared with most other fixed-income investments because they are not taxed

at the state level.

Traditionally, derivatives dealers have used LIBOR rates to proxy for the risk-free

rate. LIBOR stands for London Interbank Offered Rate , a reference rate which is

published daily to reflect the rate of interest at which banks can obtain unsecured

loans from other banks. LIBOR rates are not entirely risk-free, because there

is some probability (albeit very small) that the bank borrowing funds will default

during the life of the loan. LIBOR rates shot up during the crisis as the possi-

bility of banks defaulting on became more salient. Also, a scamdal emerged in

the wake of the crisis regarding manipulation of the reported LIBOR rates. These

What is the risk-free rate?

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issues have led many derivatives dealers to consider alternative proxies for the

risk-free rate. An increasingly popular one is the overnight indexed swap (OIS)

rate, particularly when valuing collateralized transactions.

What is the risk-free rate? (continued)

2. Forward Price for an Investment Asset

We will begin our analysis of forward prices with the simplest case: a forward contract on

an investment asset that provides no intermediate income. An example of such an asset

would be a non-dividend-paying stock or a zero coupon bond.

Consider a long forward contract to purchase a non-dividend-paying stock in 3 months.

Suppose the current price of the stock is $40 and the (annualized, continuously com-

pounded) 3-month risk-free rate is 5%. What should the forward price, F 0, of this contract

be?

To see how the forward price of the stock must relate to its current spot price, consider

two alternative ways in which an investor could lock in a price today to own the stock in

three months:

1. Take a long position in the 3-month forward contract, which costs nothing today, and

in three months pay the forward price F 0 to obtain the stock.

2. Buy the stock today for the price of $40, and hold it for three months.

Both strategies lead to ownership of the stock three months from now, and in both

cases the price to be paid for the stock is determined today. The only difference is whether

you pay now or pay later. The relationship between the spot price and the forward price

of the stock should thus be determined simply by the time value of money. Given the

3-month interest rate of 5%, paying $40 today is equivalent to paying $40 ×e5×3 / 12 =

$40.50 in three months. Therefore, the forward price for the 3-month contract should be

F 0 = $40.50.

To reinforce this point, suppose that a forward contract were available at a forward

price of $43. An arbitrageur could borrow $40 at 5% interest, buy the stock for $40, andtake a short position in the forward contract. In three months, the arbitrageur would sell

his share to the long party in the forward contract for the forward price of $43, and repay

the loan with interest, $40 ×e5%tmes3 / 12 = $40.50. The arbitrageur thus locks in a profit

of $43 - $40.50 = $2.50 at the end of three months.

A similar arbitrage opportunity would arise if the forward price were $39. In this case,

an arbitrageur would short one share of the stock for $40 and invest the proceeds at 5%

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for three months, while taking a long position in the forward contract. At the end of three

months, his $40 will have grown to $40.50. Out of this $40.50, he will purchase the stock

from the forward contract for $39 and close out the short position, retaining the difference

of $40.50 − $39 = $1.50.

In order to preclude arbitrage opportunities such as these, we see that the forwardprice must be exactly $40.50. Generalizing this argument, we have the following fact:

Fact 1. The forward price for an asset that pays no intermediate income is

F 0 = S0erT (1)

where S0 is the current spot price of the asset, T is the time in years until the delivery

date of the contract, and r is the risk-free rate for a maturity of T years.

Note that when the underlying asset pays no intermediate income, the forward priceis always greater than the spot price.

Consider a 4-month forward contract to buy a zero-coupon bond that will mature

1 year from today. The current price of the bond is $930.The continuously com-

pounded 4-month risk-free rate of interest is 6% per annum. Thus T = 4 / 12,

r = 0.06, and S0 = 930. The forward price, F 0, is

F 0,T = 930e0.06×4 / 12

= 948.79

This would be the delivery price in a contract negotiated today.

Example 1

In our analysis above, we assumed that investors can sell the asset short to take

advantage of a forward price that is too low.1 However, short sales are not possible for all

investment assets. It turns out that this does not matter, as long as there is a significant

number of people who hold the asset purely as an investment (which is how defined

investment assets in the first place). If the forward price is too low, investors who hold theasset will find it attractive to sell it and take a long position in the forward contract.

To see this, suppose that F 0 < S0erT . Then the investor who owns the asset can:

1. Sell the asset for S0.

1If the forward price is too high, an investor would take advantage of the opportunity by borrowing fundsto buy the asset and taking a short position in the forward contract. It is not necessary to short the asset inthis case.

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

3. Take a long position in a forward contract on the asset.

At time T , the cash invested will have grown to S0erT . The asset is repurchased for

F 0 under the terms of the forward contract. The investor will have the asset back in her

portfolio and will have made an arbitrage profit of S0erT − F 0.

2.1. Forward Price for an Investment Asset with Known Income

We can extend the previous analysis to a forward contract written on an asset which pays

a predictable income to the holder, such as a coupon bond or a dividend-paying stock.

Consider a long position in a 9-month forward contract to purchase a coupon bond

whose current price is $900. The bond will make a coupon payment of $40 in four months.

The 4-month and 9-month risk-free interest rates are 3% and 4%, respectively. What

should the forward price be?

As before, there are two potential strategies by which an investor can lock in a price

today to own the bond in nine months:

1. Enter the long forward contract and paying F 0 in nine months; or

2. Buying the asset today and holding it for nine months.

In the previous example, the only difference between these two strategies was whether

you pay now or pay later. In this case, there is another important difference. If you buy the

bond today and hold it, you will receive the coupon payment due in four months. If you

enter the forward contract, the coupon will have been paid before you receive the bond in

nine months.

The spot price of the bond today includes the present value of the coupon payment

that will be paid in four months, whereas at the delivery date of the forward contract, that

coupon will have already been paid out. Thus, the forward price should equal the future

value of today’s bond price, after excluding the value of the intermediate coupon payment .

Specifically, the present value of coupon payment is 40 ×e−3%×4 / 12 = $39.60, so that

the spot price of the bond excluding the value of the coupon payment is $900 - $39.60 =$860.40. Thus, the forward price should be equal to $860.40, grossed up at the 9-month

interest rate: F 0 = 860.40 × e4×9 / 12 = $886.60.

What if the forward price were too high at $910, or too low at $870? Table 6.1 summa-

rizes the trade an arbitrageur could make and the cash flows today, in four months, and

in nine months. By a no-arbitrage argument, the forward price must be equal to the spot

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Table 1: Arbitrage opportunities when 9-month forward price is out of line with spot priceof bond providing known cash income. (Bond price=$900; income of $40 occurs in 4months; 4-month and 9-month rates are 3% and 4%, respectively.)

Forward price = $910 Forward price = $870

Action now: Action now:

Borrow $900: $39.60 for 4 months Short 1 unit of bond to realize $900and $860.40 for 9 months Invest $39.60 for 4 months

Buy 1 unit of bond and $860.40 for 9 monthsEnter into forward contract to sell Enter into forward contract to buy

bond in 9 months for $910 buy in 9 months for $870

Action in 4 months: Action in 4 months: Receive $40 of income from bond Receive $40 from 4-month investmentUse $40 to repay first loan Pay income of $40 on bond

with interest

Action in 9 months: Action in 9 months:

Sell bond for $910 Receive $886.60 from 9-month investmentUse $886.60 to repay second loan Buy bond for $870

with interest Close out short position

Profit realized = $23.40 Profit realized = $16.60

price minus the present value of any intermediate income, grossed up by the risk-free

rate.

Fact 2. The forward price of an investment asset that provides intermediate income during

the life of the contract is

F 0 = (S0 − )erT (2)

where S0 is the spot price of the asset, is the present value of the intermediate income

paid by the asset during the life of the contract, T is the time in years until the delivery

date of the contract, and r is the risk-free rate for a maturity of T .

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Consider a 10-month forward contract on a stock with S0 = 50 that will pay

dividends of $0.75 per share at 3 months, 6 months, and 9 months. Suppose the

yield curve is flat at 8% interest for all maturities. Then the present value of the

dividends is

= 0.75e−0.08×3 / 12 + 0.75e−0.08×6 / 12 + 0.75e−0.08×9 / 12 = 2.162

The the forward price is then given by

F 0 = (50 − 2.162)e0.08×10 / 12 = 51.14

Example 2

2.2. Forward Price for an Investment Asset with Known Yield

Some assets, such as stock index futures or currency futures, provide income that is best

expressed as a yield rather than discrete cash payments as in the formulation above. That

is, the income is expressed as a percentage of of the asset’s price at the time that the

income is paid. Suppose that an asset provides a yield of 5% per annum. This could

mean that income accrues once per year and is equal to the 5% of the asset’s price at the

time it is paid. This would be 5% yield with annual compounding. Alternatively, incomeequal to 2.5% of the asset’s value could be paid twice a year, in which case the yield would

be 5% with semi annual compounding. We will generally express yields with continuous

compounding, in which case we can rewrite the formula for the forward price as follows:

Fact 3. Let q be the average yield per annum on an asset during the life of a forward

contract with continuous compounding. The forward price is then

F 0 = S0e(r −q)T (3)

where S0 is the current, T is the time in years until the delivery date of the contract, and

r is the risk-free rate for a maturity of T .

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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 yield

is thus 4% with semi annual compounding, which translates to a continuously

compounded yield of δ = 2× ln(1+ .02 / 2) = 3.96%. The current spot price of

the asset is $25, and the risk-free interest rate for maturity of six months is 10%.

The forward price is thus

F 0 = 25× e(0.10−0.0396)×0.5 = $25.77

Example 3

2.3. No-arbitrage bounds

In practice, trading fees, bid-ask spreads, different borrowing/lending rates, the price effectof trading in large quantities, and other market frictions make pure arbitrage trades more

difficult. In the presence of transaction costs and other frictions, we can no longer identify

an exact no-arbitrage forward price. Instead, the best we can do is identify a range within

which the forward price must fall to preclude arbitrage. That is, we can determine lower

and upper bounds F − and F + such that

F − < F 0 < F +

Consider the following trading costs and frictions:

• Bid-ask spreads: for stock Sb < S, and for forward F b < F

• Cost k of transacting forward (e.g. broker fee)

• Different interest rate for borrowing and lending such that r b < r

• For simplicity, no dividends and no time T transaction costs

Then arbitrage is possible if

1. F 0,T > F + = (S0

+ 2k )er bT , or

2. F 0,T < F − = (Sb0

+ 2k )er T

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3. TL;DR

• In deriving the relationship between spot prices and forward or futures prices, it

is convenient to distinguish between investment assets, which are held primarily

for investment purposes by a significant number of investors, and consumptionassets, which are held primarily for consumption or as inputs to production.

• For investment assets, the forward price for a contract with delivery at time T should

equal the cost of buying the asset today and financing the payment or the asset

until time T . If the forward price were higher or lower than this, it would create an

arbitrage opportunity.

– For an investment asset that provides no intermediate income (such as divi-

dend or interest payments) during the life of the forward contract, the forward

price should equal

F 0 = S0erT

or in other words, the spot price grossed up by the risk-free rate.

– For an investment asset that provides one or more predictable cash payments

during the life of the contract, the forward price is

F 0 = (S0 − )erT

where is the present value of income paid by the asset during the life of the

forward contract.

– If an asset provides income that can be expressed as a continuously com-

pounded rate or yield, the forward price can be written as

F 0 = S0e(r −q)T

where q is the annualized, continuously compounded yield on the asset.

• If there are transaction costs such as bid-ask spreads or broker fees, then instead

of a unique arbitrage-free forward price, there is a range of forward prices that can

hold without creating arbitrage opportunities.

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note5_determiningforwardprices

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