Prepaid Forward Contract

Prepaid Forward Contract A contract that delivers one unit of the underlying asset at some future date for T a price agreed upon now and payable now is a prepaid forward contract. The price you Ppay now is denoted . 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 F0,T price is just the current spot price , i. e., S P = . (1) FS0,T 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 ? Well if you buy one unit of the underlying now and reinvest the dividends, you will , ,Taccumulate units of the underlying at T. So in order to obtain one unit of the e ,,Tunderlying at date T, you need only units now. It follows that the prepaid forward e price in this context is just the price of these units today, i. e., P,,T = . (2) FSeS，，0,T Now consider a forward contract with the same underlying asset and expiration TTdate . The forward price is to be paid at date . The forward contract gives you F0,T the obligation to pay (receive) T at date in exchange for the underlying asset. How F0,T Tmuch would you pay at date to receive this obligation at date ? This is easy. You can 0 create a prepaid forward contract synthetically. Assume there is a zero coupon bond Tmaturing at date paying $1.00 face value with certainty. The price today of this zero is FI 8360. Lecture Notes. 1 Prof. D. C. Nachman denoted . Enter into a long forward contract for the asset at the forward price PT0,，， and buy zeros. Since there is no cash flow involved in entering the forward FF0,T0,T contract, the cost of this strategy is . (3) PT0,F，，0,T At date , the zeros pay you which is used to settle the long forward position and TF0,T you have the asset. The strategy initiated at date is a synthetic version of the asset, a 0 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 and note that T ,rT. Plugging this into (3) gives PTe0,,，， PrT, FF = , (4) eF，，0,0,TT0,T In a real forward contract, the forward price is determined so that both the long and F0,T short side of the contract agree on the prepaid forward price of obligation, and hence no cash has to be exchanged at date . Similarly, for any payment X determined at date 00 to be made at date T for certain, the prepaid foward price for that asset is simply its Xpresent value given by (4) with replacing . F0,T Application 1: Spot/Futures Parity (Section 22.4 in (BKM)) rT Multiplying both sides of (4) by gives the basis for the spot/futures parity e result rTP = . (5) FeF0,T0,T FI 8360. Lecture Notes. 2 Prof. D. C. Nachman 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 , continuous dividend S yield , time to expiration , riskless rate , exercise price , and volatility rTX,, involves the crucial number given by the formula d1 1,,,TrT2SeXeT，,ln，，2 = . (6) d1T, When is written in this way it is apparent that the dividend yield enters that formula d1 ,,Tonly to discount the stock price, as , and the interest rate enters the formula only to Se ,rTdiscount the exercise price, as . From (2) and (4), these values are the prepaid Xe PrTPT,,,forward prices for the stock and the exercise asset, and . FSSe,FXXe,，，，，0,T0,T The Black/Scholes formula for the Eoropean call option is ,,,TrT , (7) CSXrTSeNdXeNd(,,,,),,,，，，，12 with given in (6) and . Rewriting (6) and (7) using prepaid forward dddT,,,121 prices gives PP , (8) CSXrTFSNdFXNd(,,,,),,,，，，，，，，，0,10,2TT 1PP2FSFXT，,ln，，，，，，0,0,TT2 , (9) d,1T, FI 8360. Lecture Notes. 3 Prof. D. C. Nachman . (10) ddT,,,21 In this way, we see the important ingredients in the Black/Scholes formula are the 2prepaid forward prices (and ). We can therefore use (8)-(10) to get the value of ,T 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 as the T PrT,underlying asset. The prepaid forward price is by (4) and the value FFFe,，，0,0,0,TTT of the call option is given by (8) with and . One derives the value of the SF,,,r0,T 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. thBoadie, Z., A. Kane, and A. Marcus, 2002, Investments. 5ed. New York: McGraw- Hill/Irwin. FI 8360. Lecture Notes. 4 Prof. D. C. Nachman