(a) Consider a long forward contract to buy a non-dividend-paying stock in 3 months. Current stock price is $40 and 3-month risk-free interest rate is 5% per annum continuously compounded. If the current 3-month forward contract price is $39, explain how an arbitrageur locks in a risk-free profit and how much is the risk-free profit?

(b) A portfolio consists of a short covered call (i.e., short stock + long call) and a protective put. Assume the naked options in these trading strategies have the same underlying asset, strike, and maturity. Explain why the portfolio is equivalent to a long straddle.

(c) An investor is bearish on a stock, what are the two possible naked option strategies? Alternatively, the investor can construct a bear spread. What are the main differences between the bear spread and the naked option strategies?

(d) A portfolio consists of a long stock (i.e., current stock price is $100) and a long European put option on the stock with strike price of $105. Explain why the net payoff of portfolio is equal to or greater than $5 at maturity (Hint: ignore the premium).

(e) The 90, 100, and 110 strike calls are trading at $12, $5, $3, respectively. The stock price is at $100. Explain why the maximum net payoff is 5 on a long butterfly spread using these options.

(f) The gamma of a delta-neutral portfolio is 500. Explain why a jump of $3 in underlying asset causes the delta value to increase by 2250.