Given the stochastic differential equation for the stock price in the risk neutral measure :

dS(t) = rS(t) dt + S(t) dW (t) (1)

1. 1) Derive the value of the call option (ST K)+ = max(ST K, 0)

2. 2) Derive the value of a capped call on the stock min(max(ST K, 0), K2) , whereK2 > K

3. 3) Please describe differences between the gamma of an ATM option vs Out-of- Money option for 3 months expiry.

4. 4) Assume that you sold a 3 months at-the-money call option at volatility 25% and you hedge according to Black Scholes. The realized volatility over the 3 months turns out to be 20%. What is approximation of PNL of the delta hedged porfolio , given that the average = .03

5. 5) Consider 2 ATM options one with expiry 5 days days and one with expiry 3 months. Compare the delta , gamma and theta of the options for 5 days.