Suppose a FI has written and sold a put option to a client on 10,000 shares of a listed stock. The current share price S0 = 49, expected return = 0.05, volatility = 0.80. Moreover, the put option has a strike price of K = 50 and a maturity of 20 weeks. The FI has decided to charge the client 10% more than theoretical no-arbitrage price of the option, and then deltahedge its risk exposure on a weekly basis by trading the underlying stock and the risk-free asset. For simplicity, assume the risk-free interest rate r = 0. The FI wants to be informed about the potential profit and loss of this trade. As a result, you are required to perform the following tasks.
a. Assume the stock price follows a geometric Brownian motion, simulate 30 trajectories of weekly prices for the next 20 weeks. Plot the 30 price trajectories together in a graph.
b. For each of the price trajectory in Part a, perform a dynamic delta-hedging strategy similar to that in Table 8.2 of the textbook. Compute the profit/loss of the FI for each price trajectory
c. Plot a histogram for the profit/loss calculated in Part b. What is the average and standard deviation of the profit and loss? Estimate the probability of FI making a loss on the trade after 20 weeks.
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