(i) Following the last section, we now examine what happens if the underlying security is itself a futures contract. For example, it was seen in the last section that a call option on a stock index could be dynamically hedged by buying or selling the appropriate number of stock index futures contracts; now we consider a call option on a stock indexfutures pricerather than on the index itself.
The analysis is very similar for forward contracts and futures contracts, so these are treated together, with any divergence in behavior pointed out as we go along. Futures contracts are of course far more important in practice, since these are traded on exchanges, while active forward markets are normally interbank (especially in foreign exchange).
It is critical that the reader has a clear understanding of the concepts and notation of para- graph (iii) of the last section.
now maturity of
forward/
futures 0
time t
T t
maturity of option
(ii) The payoff of an option on a futures or t forward contract is more abstract than for a simple stock. Compare the following three European call options maturing in timeτ:
r
Options on the Underlying Stock Price:the contract is an option to buy one share of stock at a priceX.Payoff=max[(Sτ−X),0]
r
Options on the Forward Price:this is an option that at timeτwe can enter a forward contract maturing at timeT, at a forward price ofX. The value of this forward contract at timeτ will5.6 OPTIONS ON FORWARDS AND FUTURES
be (FTτ−X) e−r(τ−T).
Payoff=max
(FτT−X) e−r(T−τ),0
r
Options on the Futures Price:as in the last case, this is an option to enter a futures contract at time τand priceX; however, futures are marked to market daily so that a profit ofτT−X would immediately be realized within one day of timeτ.Payoff=max[(τT−X),0]
(iii) The forward price is given by Ft T =Ste(r−q)(T−t), and if interest rates are constant, we also haveFt T =t T. We may therefore write
Volatility ofFt T =volatility oft T =
varlnSt =σ
In general, the volatility of the forward price equals the volatility of the spot price; the volatility of the futures price equals the volatility of the underlying stock if the interest rate is constant.
(iv)Black Scholes Equation for Forwards/Futures: We shall now repeat the analysis of Sec- tion 4.2(i)–(iii), but with a forward or futures price replacing the stock price of the underlying equity stock. We use the notationVt Tto denote the forward/futurespriceandvt Tas thevalueof the contract. Using the same construction as before, we suppose that we have a small portfolio containing a forward/futures option plus units of forward/futures contracts, such that the portfolio is perfectly hedged against market movements. The value of the portfolio is
ft− tvt T = ft
The key difference between this and the previous analysis lies in this expression. For a forward contract,vt T is the value at timetof a contract to buy a unit of commodity at timeT for a price equal to the timetforward rate; but such a contract has zero value at timet. Similarly, a futures contract at timethas zero value.
Now consider an infinitesimal time intervalδt during which the forward/futures contract changes in value byδvt T. It follows from Section 5.5(iii) that
δvt T =
e−r(T−t)δFt T forward t+δt T−t T =δt T futures
Either way, we can make the undemanding assumption thatδvt T =A(Vt T,t)δVt T. The increase in value of the hedged portfolio over timetcan now be written
δft− tδvt T =δft− tAtδVt T
The arbitrage condition corresponding to equations (4.5) is δft− tAtδVt T
ft
=rδt (5.5)
It is assumed that forward and futures prices follow a similar Wiener process to a stock price:
δVt T
Vt T
=µδt+σδWt
59
5 The Black Scholes Model
Substituting this into equation (5.5) and using Ito’s lemma forδftgives ∂ft
∂t +µVt T
∂ft
∂Vt T
+1
2σ2Vt T2 ∂2ft
∂Vt T2
δt+σVt T
∂ft
∂Vt T
δWt
−Vt TAt(µδt+σδWt) t =r ftδt
The coefficient ofδWtmust equal zero, since the portfolio is perfectly hedged, so that At t = ∂ft
∂Vt T
Substituting this into the remaining terms gives
∂ft
∂t +1
2σ2Vt T2 ∂2ft
∂Vt T2
=r ft (5.6)
(v)Significance of the Simplified Black Scholes Equation:The equation which has just been derived holds for forward prices and for futures prices. In the case of futures contracts, it does not depend on the idealized assumptions which were used to equate the forward and futures prices, i.e. constant interest rates.
The equation is simpler than the Black Scholes equation for options on an equity stock. The reason can be traced to equation (5.5): the cost of entering a forward or futures contract is zero, and these instruments have no dividend throw-off. Consequently, the financing costs for the hedge are zero and the financing term reduces merely to the cost of carrying the option itself.
This becomes immediately plain by examining the Black Scholes equation written in the form of equation (4.20).
The partial differential equation for forwards/futures has the same form as the general Black Scholes equation for an equity stock, in which one has setq =r.This is in line with the properties of forwards and futures with which we are already familiar. Consider first the forward price: from equation (3.4) we have
Et[ST]risk neutral=Ste(r−q)(T−t)=Ft T
where the symbol Et[·]risk neutralindicates that the expectation is taken at timetandrisk neutral means that we have setµ→r. Then
Et[FτT]risk neutral =Et
Sτe(r−q)(T−τ)
risk neutral = e(r−q)(T−τ)Et[Sτ]risk neutral
=e(r−q)(T−τ)Ste(r−q)(τ−t)=Ft T
The risk-neutral expected growth rate of Ft T is therefore zero, which is the same as for an equity whereq =r.
Clearly, this same result would hold for a futures price whent T =Ft T, i.e. when interest rates are constant. However the result is more general, and holds for variable interest rates also. The reason is that a futures contract costs nothing to enter so that arbitrage assures that the expected profit from the contract must be zero:
Et[τT]risk neutral=t T
(vi)Black ’76 Model:We have established that the Black Scholes equation for an option on a forward/futures price can be obtained from the general equation for an option on the equity
5.6 OPTIONS ON FORWARDS AND FUTURES
price by settingq →r; therefore, the Black Scholes formula for an option on a forward or futures price can be obtained from the general Black Scholes formula by just the same procedure:
ft = e−r(τ−t)φ{Vt TN[φd1]−XN[φd2]} d1= 1
σ√ τ −t
lnVt T
X +1
2σ2(τ−t)
; d2=d1−σ√
τ−t; Vt T =Ft T or t T
(5.7)
61
6
American Options
Apart from a couple of sections in Chapter 2 and a few cursory references elsewhere, this book has so far concentrated on the behavior of European options. These are relatively easy to value and formulas normally exist for calculating prices and hedge parameters; but all of the general option theory that was developed in Chapter 4 applies toanyderivative which may be perfectly hedged with underlying stock – American as well as European. In view of the widespread use of American options, especially in the exchange traded markets, the pricing of these will now be examined.
It should be stated at the outset that although the material in this chapter throws light on the nature of American options, the most common ways of evaluating these options are dealt with in later chapters. The reader in a hurry may prefer to limit his attention to Section 6.1.