A compound option is an option on an option. Suppose there is a constant risk-free rate, and the underlying asset price has a constant volatility. Consider a European call option with strike K maturing at T

. Consider now a European option maturing atT

to purchasethe first call option at priceK. The purpose of this exercise is to value the “call on a call.” The value at T of the underlying call is given by the Black-Scholes formula with T − T being the time to maturity.

Denote this value by V(T,ST). The value at T of the call on a call is max(0,V(T,ST) −K).

Let S∗ denote the value of the underlying asset price such that V(T,S∗) = K. The call on a call is in the money at its maturity T if ST > S∗, and it is out of the money otherwise. Let A denote the set of states of the world such that ST > S∗, and let 1A denote the random variable that equals 1 when ST > S∗ and 0 otherwise. The value at its maturity T of the call on a call is V(T,ST)1A − K1A .

(a) What is the value at date 0 of receiving the payoff K1A at date T?

(b) To value receiving V(T,ST)1A at date T, let C denote the set of states of the world such that ST > K

, and let 1C denote the random variable that equals 1 when ST > K and 0 otherwise. Recall that V(T,ST) is the value at T of receiving ST1C −K

1C at date T

. Hence, the value at date 0 of receiving V(T,ST)1A at date T must be the value at date 0 of receiving (ST1C − K

1C)1A at date T

.

(i) Show that the value at date 0 of receiving V(T,ST)1A at date T is S0 probS

(D) −K

e−rT

probR(D), where D = A∩C.

(ii) Show that probS

(D) is the probability that

− B∗

T √T

< d1 and − B∗

T

√T

< d

1 , where d1 = log(S0/S∗)+ 

r + 1 2σ2



T

σ

√T , d

1 = log(S0/K

) +

r + 1 2σ2



T

σ

√T , and where B∗ denotes a Brownian motion under probS

. Note that the random variables in (16.32) are standard normals under probS with a correlation equal to √T/T

. Therefore, probS

(D) can be computed from the bivariate normal cumulative distribution function.

(iii) Show that probR(D)is the probability that

− B∗

T √T

< d2 and − B∗

T

√T < d

2 , (16.33)

where d2 = d1 − σ

√

T, d

2 = d

1 − σ

√

T , and where B∗ now denotes a Brownian motion under probR.