Black-Scholes-Merton pricing formulas Suppose that So is the stock price at time zero, K is the strike price, r is the continuously compounded risk- free rate, o is the stock price volatility, and T is the time to maturity of the option. The European call price c and the European put price p are c = SoN(d) Ke='T N(d) and p = Ke='T N(-d2) SoN(-d), respectively, where In(So/K) + (1 + 02/2) T da = Inc In(So/K) + (1 - 04/2)T = di-OVT, di = ONT t = d - OVT, and the function N(x) is the cumulative probability distribution function for a standardized normal distribu- tion. (4) With the notation used in the above formulas: (a) What is N'(x)? (b) Show that SN'(di) = Ke-r(Tt)N'(d2), where S is the stock price at time t and , In(S/K) + (1 + 02/2) (T 1) di _In(S/K) + (1 - 02/2) (T t) OVT - 7 OVT - 7 (c) Calculate di/aS and 8d2/as. (d) Show that when C = SN(d) Ke-r(T-t) N(d2) d = it follows that =-rKe-r(T-1) N(d2) SN'(d) 2T - t where c is the price of a call option on a non-dividend-paying stock. (e) Show that dc/as = N(di). (f) Show that c satisfies the Black-Scholes-Merton differential equation. (g) Show that c satisfies the boundary condition for a European call option, i.e., that c = max(S K,0) as t + T. Black-Scholes-Merton pricing formulas Suppose that So is the stock price at time zero, K is the strike price, r is the continuously compounded risk- free rate, o is the stock price volatility, and T is the time to maturity of the option. The European call price c and the European put price p are c = SoN(d) Ke='T N(d) and p = Ke='T N(-d2) SoN(-d), respectively, where In(So/K) + (1 + 02/2) T da = Inc In(So/K) + (1 - 04/2)T = di-OVT, di = ONT t = d - OVT, and the function N(x) is the cumulative probability distribution function for a standardized normal distribu- tion. (4) With the notation used in the above formulas: (a) What is N'(x)? (b) Show that SN'(di) = Ke-r(Tt)N'(d2), where S is the stock price at time t and , In(S/K) + (1 + 02/2) (T 1) di _In(S/K) + (1 - 02/2) (T t) OVT - 7 OVT - 7 (c) Calculate di/aS and 8d2/as. (d) Show that when C = SN(d) Ke-r(T-t) N(d2) d = it follows that =-rKe-r(T-1) N(d2) SN'(d) 2T - t where c is the price of a call option on a non-dividend-paying stock. (e) Show that dc/as = N(di). (f) Show that c satisfies the Black-Scholes-Merton differential equation. (g) Show that c satisfies the boundary condition for a European call option, i.e., that c = max(S K,0) as t + T