Exercise 2. Consider the standard Black-Scholes model and a T-claim X of the form X (S(T)). Denote the corresponding arbitrage free price process by II(t). = (a) Show that, under the martingale measure Q, II(t) has a local rate of return equal to the short rate of interest r. In other words show that II(t) has a differential of the form dII(t) =r. II(t)dt + g(t)dW(t) Hint: Use the Q-dynamics of S together with the fact that F satisfies the pricing PDE. - (b) Show that, under the martingale measure Q, the process Z(t) = II(0) B(t) is a martingale. More precisely, show that the stochastic differential for Z has zero drift term, i.e. it is of the form dz(t) = Z(t)oz(t)dW(t) = Determine also the diffusion process oz(t) in terms of the pricing func- tion F and its derivatives. Exercise 2. Consider the standard Black-Scholes model and a T-claim X of the form X (S(T)). Denote the corresponding arbitrage free price process by II(t). = (a) Show that, under the martingale measure Q, II(t) has a local rate of return equal to the short rate of interest r. In other words show that II(t) has a differential of the form dII(t) =r. II(t)dt + g(t)dW(t) Hint: Use the Q-dynamics of S together with the fact that F satisfies the pricing PDE. - (b) Show that, under the martingale measure Q, the process Z(t) = II(0) B(t) is a martingale. More precisely, show that the stochastic differential for Z has zero drift term, i.e. it is of the form dz(t) = Z(t)oz(t)dW(t) = Determine also the diffusion process oz(t) in terms of the pricing func- tion F and its derivatives