Problem 1 (Delta hedging) The purpose of this exercise is to use simulation in python / R to measure the hedging error resulting from discrete rebalancing of a hedge. You sell a 3-month European call option at the Black-Scholes price and try to hedge it by holding delta shares of the underlying stock. You can borrow money from the bank at constant interest rate, and any money left in your account earns the same constant rate of interest.

- At initiation of the contract, you get the option premium from the client and buy delta shares of the stock. You may need to borrow extra money to set up your hedging portfolio.

- At each time step, the stock price has evolved from the previous step and the delta must be adjusted. Depending on how it has changed, you need either to buy or sell shares. You also pay or earn interest on any money borrowed or deposited over the previous period.

- At maturity, you close your position. This means selling all shares you own, reimbursing the bank for the money you owe or get what is left in your account, and paying your client the amount max(0, ST K). How much cash is left after that is your profit or loss.

Assume the underlying asset S is modeled by a geometric Brownian motion and use the parameters below. BlackScholes theory says that, as the number of times you rebalance goes to infinity (with T fixed), your hedging error (profit or loss) goes to zero on each path. Since, in practice, continuous trading is impossible, the hedge is imperfect and we want to study this imperfection.

Initial price: S0 = 50, Rate of return: = 10% Volatility: = 30%, Interest rate: r = 5% Strike: K = 50, Expiration: T = 0.25

(a) Find the mean and standard deviation of the hedging error with daily and weekly rebalancing. Make a histogram of the distribution of hedging errors for both cases.

(b) Take different values for . How does that change the results? Explain your findings.

(c) Let t denote the rebalancing interval. We know that as t 0 the hedging error goes to 0. What does your simulation suggest about the rate of convergence? Does the hedging error appear to be O((t) ) for some , and if so, what ? (Hint: You may want to draw a log-log plot.)