Q 4. We denote by r > 0 the risk-free interest rate. Recall the Black-Scholes model and the Black Scholes formula for a T-expiry, K-strike European call option written on S having positive constant instantaneous volatil- a ity e [0, T) (S, t, K, T, r, o) SN(d) "("-) N(d,), '. Here, N() is the cumulative distribution function of the standard normal distribution, i.e 1 -v2 /2 dv N(x) _ V27T and In(Se(T-)/K) d1,2 oVT-t t _ 2 You are to verify that S (S, t, K, T, r, ) C(S, t) satisfies the Black-Scholes PDE with the appropriate boundary condition (a) Compute C/at, aC/as, and C/as2 3 marks (b) Verify that C(S,t) satisfies 10 C 2aS20 rSi at (S,t) e (0, o) [0, ) . rC, X as 3 marks (S- K) 3 marks (c) Verify that for each fixed S, we have the limit C(S, t) as t T Q 4. We denote by r > 0 the risk-free interest rate. Recall the Black-Scholes model and the Black Scholes formula for a T-expiry, K-strike European call option written on S having positive constant instantaneous volatil- a ity e [0, T) (S, t, K, T, r, o) SN(d) "("-) N(d,), '. Here, N() is the cumulative distribution function of the standard normal distribution, i.e 1 -v2 /2 dv N(x) _ V27T and In(Se(T-)/K) d1,2 oVT-t t _ 2 You are to verify that S (S, t, K, T, r, ) C(S, t) satisfies the Black-Scholes PDE with the appropriate boundary condition (a) Compute C/at, aC/as, and C/as2 3 marks (b) Verify that C(S,t) satisfies 10 C 2aS20 rSi at (S,t) e (0, o) [0, ) . rC, X as 3 marks (S- K) 3 marks (c) Verify that for each fixed S, we have the limit C(S, t) as t T