An option on an option, sometimes called a compound option, is specified by the parameter pairs (K1, t1) and (K, t), where t1 < t. The holder of such a compound option has the right to purchase, for the amount K1, a (K, t) call option on a specified security. This op- tion to purchase the (K,t) call option can be exercised any time up to time t1.
(a) Argue that the option to purchase the (K, t) call option would never be exercised before its expiration time t1.
(b) Argue that the option to purchase the (K,t) call option should be exercised if and only if S(t1) ≥ x, where x is the solution of K1 =C(x,t−t1,K,σ,r), C(s, t, K, σ, r) is the Black-Scholes formula, and S(t1) is the price of the security at time t1.
(c) Argue that there is a unique value x that satisfies the preceding identity.
(d) Argue that the unique no-arbitrage cost of this compound option can be expressed as no-arbitrage cost of compound option
where: s = S(0) is the initial price of the security; x is the value specified in part (b); W is a normal random variable with mean (r − σ2/2)t1 and variance σ2t1; I(seW > x) is defined to equal 1 if seW > x and to equal 0 otherwise; and C(s,t,K,σ,r) is the Black-Scholes formula. (The no-arbitrage cost can be simplified to an expression involving bivariate normal probabilities.)
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