8. Suppose that the stock price follows a jump-diffusion process. 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).