a. A stock price follows a geometric Brownian motion with an expected return of 16% and a volatility of 34%. The current price is $38.

i. What is the real-world probability that a European call option on the stock, with an exercise price of $40 and a maturity date in six months, will be exercised?

ii. Assume the risk-free rate is 5%, what is the risk-neutral probability that a European call option on the stock, with an exercise price of $40 and a maturity date in six months, will be exercised?

b. Assume that a non-dividend-paying stock has an expected return and volatility , and the risk-free rate is r per annum. Show that the value of a long forward contract, given by ft = St K e-r(T-t), satisfies the Black-Scholes-Merton differential equation, and the boundary condition, f = S K at maturity when t = T.