The black-scholes formula by change of numeraire let st be

Question

The Black-Scholes formula by change of numeraire Let St be the price at time t of a stock that pays no dividends.

(a) Draw the payout functions for the two options

(i) KI{ST ≥ K}

(ii) STI{ST ≥ K}

where I{} is the indicator function. Which linear combination of (i) and (ii) gives the payout of a call option?

(b) Suppose the risk-neutral distributions of S_T conditional on S_t with respect to the money market and stock numeraires are given by the answers to Question 5(c) and (e), respectively. Price options (i) and (ii) in terms of the normal cumulative distribution function Φ (·) using the money market numeraire for (i) and the stock numeraire for (ii). Hence derive immediately the Black-Scholes equation for the price of a call option.

Question 5(c)

(c) Suppose the risk-neutral distribution of S_T conditional on S_t, with respect to the money market numeraire, is lognormal (log S_t + ν(T - t), σ²(T - t)), that is, log S_T | S_t ∼ N(log S_t + ν(T - t), σ²(T - t)). Use a martingale condition for S_T/M_T to find ν in terms of σ and r, the constant, continuously compounded interest rate.