In this problem we will use Monte Carlo to simulate the behavior of the martingale St/Pt, with Pt as numeraire. Let x0 = S0/P0(0, T). Simulate the process
xt+h= (1+ σ√hZt+h)xt
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. Did you simulate a martingale?
c. Verify that the result is approximately the same as the price of a cash-or nothing call computed as e−rT N(d2) ($0.5766 for the above parameters).