Prepaid Forward Contract

A contract that delivers one unit of the underlying asset at some future date T for a price agreed upon now and payable now is a prepaid forward contract. The price you

P pay now is denoted F0,T . The corresponding forward price for the underlying is denoted

F0,T . If the underlying asset has zero yield, it is easy to argue that the prepaid forward price is just the current spot price S , i. e.,

P F0,T = S . (1)

One way to obtain one unit at T and pay for it now is just to buy it now and hold till T .

With no yield, owning the asset now or paying for it now and getting it at date T are not

(financially) different.

What if the underlying asset pays a continuously compounded dividend yield of d ? Well if you buy one unit of the underlying now and reinvest the dividends, you will accumulate edT units of the underlying at T . So in order to obtain one unit of the underlying at date T , you need only e-dT units now. It follows that the prepaid forward price in this context is just the price of these units today, i. e.,

P -dT F0,T ( S ) = e S . (2)

Now consider a forward contract with the same underlying asset and expiration

date T . The forward price is F0,T to be paid at date T . The forward contract gives you

the obligation to pay (receive) F0,T at date T in exchange for the underlying asset. How much would you pay at date 0 to receive this obligation at date T ? This is easy. You can create a prepaid forward contract synthetically. Assume there is a zero coupon bond

FI 8360. Lecture Notes. 1 Prof. D. C. Nachman maturing at date T paying $1.00 face value with certainty. The price today of this zero is denoted P(0, T ) . Enter into a long forward contract for the asset at the forward price

F0,T and buy F0,T zeros. Since there is no cash flow involved in entering the forward contract, the cost of this strategy is

P(0, T ) F0,T . (3)

At date T , the zeros pay you F0,T which is used to settle the long forward position and you have the asset. The strategy initiated at date 0 is a synthetic version of the asset, a prepaid forward contract and (1) gives the prepaid forward price for the asset.

Thus the prepaid forward price is just the present value of the forward price F0,T .

Let r be the continuously compounded riskless interest rate for date T and note that

P(0, T) = e-rT . Plugging this into (3) gives

P -rT F0,T( F 0, T ) = e F0,T , (4)

In a real forward contract, the forward price F0,T is determined so that both the long and short side of the contract agree on the prepaid forward price of obligation, and hence no cash has to be exchanged at date 0 . Similarly, for any payment X determined at date 0 to be made at date T for certain, the prepaid foward price for that asset is simply its

present value given by (4) with X replacing F0,T .

Application 1: Spot/Futures Parity (Section 22.4 in (BKM))

Multiplying both sides of (4) by erT gives the basis for the spot/futures parity result

FI 8360. Lecture Notes. 2 Prof. D. C. Nachman rT P F0,T = e F0,T . (5)

Plugging in from (1) for the prepaid forward price on an asset with no dividends gives the spot futures parity result for non-dividend paying stocks. Plugging in from (2) for the prepaid forward price for an asset with a continuously compounded dividend yield gives the spot/futures parity result for this case.

Application 2: Reinterpretation of Black/Scholes.

The Black/Scholes formula for a stock with current price S , continuous dividend yield d , time to expiration T , riskless rate r , exercise price X , and volatility s

involves the crucial number d1 given by the formula

1 ln (Se-dT Xe - rT ) + s 2 T d1 = 2 . (6) s T

When d1 is written in this way it is apparent that the dividend yield enters that formula only to discount the stock price, as Se-dT , and the interest rate enters the formula only to discount the exercise price, as Xe-rT . From (2) and (4), these values are the prepaid

P-d T P- rT forward prices for the stock and the exercise asset, F0,T ( S) = Se and F0,T ( X) = Xe .

The Black/Scholes formula for the Eoropean call option is

-dT - rT C( S , X , r , T ,s ) = Se N( d1) - Xe N( d 2 ) , (7)

with d1 given in (6) and d2= d 1 -s T . Rewriting (6) and (7) using prepaid forward prices gives

P P C( S , X , r , T ,s ) = F0,T( S) N( d 1) - F 0, T ( X) N( d 2 ) , (8)

FI 8360. Lecture Notes. 3 Prof. D. C. Nachman P P 1 2 ln (F0,T( S) F 0, T ( X)) + s T d = 2 , (9) 1 s T

d2= d 1 -s T . (10)

In this way, we see the important ingredients in the Black/Scholes formula are the prepaid forward prices (and s 2T ). We can therefore use (8)-(10) to get the value of

European call options on an asset for which we can determine these prepaid forward prices.

As one application, consider a forward contract that matures at T as the

P- rT underlying asset. The prepaid forward price is by (4) F0,T( F 0, T) = F 0, T e and the value

of the call option is given by (8) with S= F0,T and d = r . One derives the value of the corresponding European puts by put/call parity (Section 20.4 in (BKM)). This is the

Black (1976) model for futures options. It is used quite regularly in industry.

References:

McDonald, R. L., 2003. Derivatives Markets. Boston: Addison Wesley. The material here was taken from chapters 5 and 12. See in particular section 12.2 for the other extensions to Black/Sholes.

Black, F., 1976. "The Pricing of Commodity Contracts." Journal of Financial Economics

3, 167-179.

FI 8360. Lecture Notes. 4 Prof. D. C. Nachman Boadie, Z., A. Kane, and A. Marcus, 2002, Investments. 5thed. New York: McGraw-

Hill/Irwin.

FI 8360. Lecture Notes. 5 Prof. D. C. Nachman