1. Your first task is to provide an analysis of the effectiveness of dynamic hedging of an options position.

a. A trader sells 100,000 European call options on a non-dividend paying stock when stock price is $49, strike price is $50, risk-free interest rate is 5% p.a., stock price volatility is 50% p.a., and time to maturity is one year. The trader wants to charge 25% more than the no-arbitrage price. How much should the trader charge? Note: the no-arbitrage price in this case is the Black-Scholes price. (1 mark)

b. Assume the stock price follows a geometric Brownian motion with expected return, = 0, and volatility = 0.5, simulate two price trajectories using {( ) } 2 exp 0.5 tt t S S t t + = + , where is a standard normal random variable, t=1/252 (daily increment), and time horizon is one year. For the first price trajectory, we want the price at maturity to be less than the strike price, i.e., ST < K, such that the call option closes out-of-themoney. For the second price trajectory, we want the price at maturity to be greater than the strike price, i.e., ST > K, such that the call option closes in-the-money. Plot the two price trajectories on the same graph. Note: since the price trajectories are randomly generated, you may need to repeat the simulation multiple times until you find one that fits the criterion. (2 marks)

c. For each of the price trajectory in Part (b), perform delta-hedging similar to Table 8.2 and Table 8.3 on pp.167-168 of the textbook. What are the hedging costs for the case in which option closes out-of-the-money, and for the case in which option closes in-the-money, respectively? (5 marks)