Consider Tutorial question 14.22, where a bank has written a European call option on one stock and a European put option on another stock. Instead of using linear approximation, the bank wants a more accurate estimate of the 10-day 99% VaR by using Monte-Carlo simulations.

a. Generate 5000 simulated prices in ten days for each of the two stocks. Assume the expected returns are zero.

b. Based on the simulated stock prices, compute the percentage return for the bank's position in the options for each simulation trial. Report the sample mean, standard deviation, and the 10-day 99% VaR for the bank's position (all should be in percentage returns). What is the Sharpe ratio of the bank's position? Why do you think Sharpe ratio might not be an appropriate performance measure in this case?

c. Draw a histogram of the simulated 10-day portfolio returns. You should plot the percentage returns on the x-axis and probability density on the y-axis. On the same graph, plot the probability density function of the normal distribution based on the delta-approximation used in Tutorial question 14.22.

d. Comment on the difference in the distributions in part (c). Which distribution leads to a higher value-at-risk? Why?

14.22. A bank has written a European call option on one stock and a European put option on another stock. For the first option, the stock price is 50, the strike price is 51, the volatility is 28% per annum, and the time to maturity is nine months. For the second option, the stock price is 20, the strike price is 19, the volatility is 25% per annum, and the time to maturity is one year. Neither stock pays a dividend, the risk-free rate is 6% per annum, and the correlation between stock price returns is 0.4. Calculate a 10-day 99% VaR.