Consider a non-dividend-paying stock whose current price S(0) = S is $50. After each period, there is a 40% chance that the stock price goes up by 25%. If the stock price does not go up, then it drops by 20%. A European call option and a European put option on this stock expire on the same day in 4 months at $54 strike. Current risk-free interest rate is 9% per annum, compounded continuously.

(a) Use the Black-Scholes option pricing formula to calculate the call option price after four months. Use σ= ln u/sqrt(∆t) with u is the up-factor and ∆t = 1/12

(b) Find the Delta, Theta, and Gamma of the option at time 0.

(c) After a week the stock price has gone down to $48. Use the result of part (b) to find the approximate value of the option at the end of week 1. Assume there are 52 weeks in a year.

(d) Use the Black-Scholes put option pricing formula K*e^{−r(T−t)}*N(−d2) − S*N(−d1) to calculate the put option price for case of (c), i.e. when the stock price is $48 at week 1.

(e) Use put-call parity and the result of (d) to derive the actual call price in (c), then compare it with the result of (c).