t T
t 0
now maturity of
underlying option time t maturity
of compound option
(i)Definitions:We will consider an option (thecom- poundoption) on anunderlyingoption. Both the compound and the underlying options can be either put or call options, so that we have four options to consider in all. Half the battle in pricing these
options is simply getting the notation straight, and this can be summarized as follows:
UNDERLYING STOCK:
Standσ Stock price at timetand volatility UNDERLYING OPTIONS:
Cu(St,X,t);Pu(St,X,t) Value at timetof an underlying call/put option. The general case is writtenU(St,X,t)
T;X Maturity date and strike of underlying options COMPOUND OPTIONS:
CC;PC;CP;PP Value at timetof a call on a call, put on a call, etc. The general case is writtenU(St,K,t)
τ;K Maturity date and strike of the compound options
Sτ∗ Critical stock price at timeτ, which determines whether or not the compound option is exercised. It is the value ofSτ that solves the equationK =u(Sτ∗,X, τ)
14 Options on One Asset at Two Points in Time
(ii)Payoffs of Compound Options:Before moving on to pricing formulas, it is worth getting an idea of the likely shape of the curve of the compound price. Let us start with a call on a call. At timeτ, the price of the underlying call is given by the curve shown in Figure 14.1. The payoff of the compound call option is defined as
CC(τ)=max[Cu(τ)−K,0]
DefineSτ∗as the value ofSτfor whichK =CU(Sτ∗,X, τ). Clearly, the payoff diagram is made up of thex-axis and that part of the curveCU(τ) which lies aboveK. This is shown as the solid line in Figure 14.2, together with the compound option price before maturity (dotted curve).
St
*t
S X e-r (T-t) K
CU(t)
Figure 14.1 Underlying option (call)
St t*
S
CU(t) C ( t )U
Figure 14.2 Compound option (call on call) Using the same analysis as for the call on call above, a put option on the underlying stock is represented by the curve shown in Figure 14.3. The payoff of the compound put option, shown in Figure 14.4, is
PP(τ)=max[K−Pu(τ),0]
PU(t)
St
*t
S X e-r (T-t) K
Figure 14.3 Underlying option (put)
PU(t)
St
*t
S CP(t)
Figure 14.4 Compound option (put on put) The remaining two compound options have curves shown in Figures 14.5 and 14.6.
(iii) Consider the most common case: a call on a call. In order to calculate the value of this compound option, we use our well-established methodology of finding the expected value of the payoff in a risk-neutral world, and discounting to present value at the risk-free rate; but now, the ultimate payoff is a function of two future stock prices (Geske, 1979):
14.1 OPTIONS ON OPTIONS (COMPOUND OPTIONS)
St
CP(t)
Figure 14.5 Compound option (call on put)
St
PC(t)
Figure 14.6 Compound option (put on call)
r
Sτ –The stock price when the compound option matures. If this is less than some critical valueSτ∗, it will not be worth exercising the compound option since the underlying option would then be cheaper to buy in the market. This may be writtenonly exercise if Sτ∗ <Sτ,where Sτ∗ is the solution to the equation K =S∗τe−q(T−τ)N[d1∗]−Xe−r(T−τ)N[d2∗]
d1∗ andd2∗are the usual Black Scholes parameters with the stock price set equal toSτ∗.
r
ST –The stock price when the underlying option matures. The ultimate payoff at timeTis the payoff of the underlying call, if (and only if) the conditionSτ∗<Sτ was fulfilled. This ultimate payoff is of course a function ofST.Since the value of a compound option depends on the expected values of bothSτ andST, we must examine theirjointprobability distribution.
(iv) Following the approach of Section 5.2(i) for the Black Scholes formula, the price of this option may be expressed as
CC(0)=PV
E
payoff of underlying option
−payment for underlying optiononly if compound option was exercised
risk neutral
r
“Only if compound option exercised”≡Sτ∗<Sτ.r
“Payment for underlying”=K at timeτ.r
“Payoff of underlying option” (at timeT)=max[0,ST−X].Combining this together and simplifying the notation gives
CC(0)= e−r T E[max[ST −X,0]:Sτ∗<Sτ]− e−rτE[K:Sτ∗<Sτ]
= e−r TE[ST−X:S∗τ <Sτ;X <ST]− e−rτKP[Sτ∗<Sτ] (14.1) These expectations are evaluated explicitly in Appendix A.1(v) and (ix), to give
CC(0)=S0e−q T N2[d1,b1;ρ]−Xe−r T N2[d2,b2;ρ]−Ke−rτ N[b2] (14.2a) 171
14 Options on One Asset at Two Points in Time
where
d1= 1 σ√
T
lnS0e−q T Xe−r T +1
2σ2T
; b1= 1
σ√ τ
lnS0e−qτ Sτ∗e−rτ +1
2σ2τ
d2=d1−σ√
T; b2=b1−σ√
τ; ρ = τ
T
and Sτ∗ is a solution to the equation K =CU(Sτ∗,X, τ). The lower limits of integration in the above-mentioned appendices are the values ofzτ andzT corresponding to Sτ=Sτ∗ and ST =X.
(v)General Formula:The put–call parity relationship
CC(0)+Ke−rτ =PC(0)+CU(0)
may be used to calculate the formula for a put on an underlying call. The relationships N[d]+N[−d]=1 and N2[d,b;ρ]+N2[d,−b;−ρ]=N[d] [see equation (A1.17)] are used to simplify the algebra, giving
PC(0)=Xe−r TN2[d2,−b2;−ρ]−S0e−q TN2[d1,−b1;−ρ]+Ke−rτN[−b2] Similar results are obtained for put and call options on an underlying put option. The four possibilities for compound options can be summarized in the general formula
U(0)=φUφ{S0e−q TN2[φUd1, φUφb1;φρ]−Xe−r TN2[φUd2, φUφb2;φρ]}
−φKe−rτN[φUφb2] (14.2b)
where
d1 = 1 σ√
T
lnS0e−q T Xe−r T +1
2σ2T
; b1= 1
σ√ τ
ln S0e−qτ Sτ∗e−rτ +1
2σ2τ
d2=d1−σ√
T; b2 =b1−σ√
τ; ρ= τ
T; Sτ∗ solves K =U(Sτ∗,X, τ) φU =
+1 underlying call
−1 underlying put φ =
+1 compound call
−1 compound put
(vi)Installment Options:When they are first encountered, compound options often look to stu- dents like rather contrived exercises in option theory. However they do have very practical applications, as the following product description indicates:
r
An investor receives a European call option which expires at timeTand has strikeX.r
Instead of paying the entire premium now, the investor pays a first installment ofCCtoday.r
At timeτ, the investor has the choice of walking away from the deal or paying a second installmentKand continuing to hold the option.This structure clearly has appeal in certain circumstances; it is just the call on a call described in this section, but couched in slightly less dry terms.