Question 1 (The Black-Scholes formula). a) Suppose for t

Question 1 (The Black-Scholes formula). a) Suppose for t T, a stock that pays no dividends has risk-neutral distribution STISt given by 1 log STISt N (v, 02 (T t)), where v log St + r 02 (T t) 2 and r is the continuous interest rate and ff is the lognormal volatility. Show that the price at time t of a K-strike call with exercise date T is given by CK(t, T) Z(t, T) (F(t, 10(d2)) , where log(St/K) + (r + (T t) dl = Hints: i) Let ST = eYT where YT is normally distributed. ii) Be careful about the range of integration of Y. iii) Use the identity 2027 1 202T = dl - ovq=-t. where T = T t. b) Use put-call parity to show that the price PK(t, T) of a European put is given by PK(t, T) = Z(t, - F(t, T)q-dl)). You should assume that the stock pays no dividends. Hint: Use your answer from part (a) and the fact that Question 1 (The Black-Scholes formula). a) Suppose for t T, a stock that pays no dividends has risk-neutral distribution STISt given by 1 log STISt N (v, 02 (T t)), where v log St + r 02 (T t) 2 and r is the continuous interest rate and ff is the lognormal volatility. Show that the price at time t of a K-strike call with exercise date T is given by CK(t, T) Z(t, T) (F(t, 10(d2)) , where log(St/K) + (r + (T t) dl = Hints: i) Let ST = eYT where YT is normally distributed. ii) Be careful about the range of integration of Y. iii) Use the identity 2027 1 202T = dl - ovq=-t. where T = T t. b) Use put-call parity to show that the price PK(t, T) of a European put is given by PK(t, T) = Z(t, - F(t, T)q-dl)). You should assume that the stock pays no dividends. Hint: Use your answer from part (a) and the fact that