An at-the-money (S0 = X) 2-month European call option is on a non-dividend-paying stock. The risk-free interest rate is 5% per annum and the stock return volatility is 30% per annum. Stock price is currently $100.

A. What is the Delta and Gamma of the option, according to the Black-Scholes Model?

(b) A dealer has a short position in 1000 call options, and would like to have a delta- neutral portfolio. How many shares should the dealer buy? Assume fractions of shares are available.

c. Now the dealer has established the positions of calls and stocks in (b), what are the values of the call position, stock position and total position respectively, according to the Black- Scholes Model?

(d) Now the dealer has established the positions of calls and stocks in (b), but the stock price suddenly jumps to $98. What is the total value of the positions that consist of the calls and shares, according to the Black-Scholes Model?

(e) In order to maintain an approximate delta-neutral portfolio when the stock price suddenly jumps to $98,how many shares should the dealer buy or sell additionally, relative to the positions in (b), according to the Gamma rule? Assume fractions of shares are available.