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1. 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?
2. Your second task is to estimate VaR for an equity portfolio using the “Model-Building” approach. Suppose you have invested in two of the Fama-French factors, namely Mkt-RF and HML. Assume conditionally the returns of Mkt-RF and HML follow a bivariate normal distribution. Assume the portfolio return is the sum of the factor returns.
a. Estimate the daily volatilities and the correlation using the EWMA model with λ = 0.94. Plot the daily volatilities for Mkt-RF and HML on the same graph. Plot the daily correlations on a separate graph. Estimate the one-day 99% VaR for the portfolio. Note: You are not required to use maximum likelihood to estimate λ, we simply assume λ = 0.94. You may assume the initial variance for the first rate of return is equal to the sample variance, and the initial covariance is equal to the sample covariance.
b. Back-test the VaR model in Part (a) using the Kupiec two-tailed test at the 5 percent significance level, and at the 1 percent significance level. Write down your conclusion.
3. Based on the methodology described in Section 11.6.2 on p.262 of the textbook, estimate the unconditional default rate, PD, and the default correlation ρ using the annual percentage default rates from 2002-2016 in Table 11.6. Plot the probability distribution of default rate.
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