Suppose that the stock price follows a jump-diffusion process as outlined in Section 20.7. Let the jump intensity be λ = 0.75, the expected jump exp(αJ ), with αJ
=
−0.15, and let the jump volatility be σJ
= 0.25. You can simulate the behavior of the martingale St/Pt as xt+h= [1− λkh + σ√hZt+h+ J(Y − 1)] xt
where k = exp(αJ ) − 1, J = 1 indicates a jump and J = 0 otherwise, and Y = eαJ
−0.5σ2
J +σJWt , with Wt standard normal. Let h be approximately 1 day.
a. Evaluate P0E_ST /PT (T , T) > K_.
b. Compute the mean and standard deviation of the difference xT− x0. Verify that you have simulated a martingale.
c. Verify that the result is approximately the same as the price of a cash ornothing call ($0.5865 for the above parameters).