Consider the daily returns of SPX in d-aaspx9808.txt. Transform the returns into \(\log\) returns and focus on the daily negative log returns.

(a) Fit the generalized extreme value distribution to the negative SPX log returns, in percentage, with subperiods of 21 trading days. Write down the parameter estimates and their standard errors. Obtain a scatterplot and a QQ plot of the residuals.

(b) What is the return level of the prior fitted model when 24 subperiods of 21 days are used?

(c) Obtain a QQ plot (against exponential distribution) of the negative log returns with threshold \(2.5 \%\) and a mean excess plot of the returns.

(d) Fit a generalize Pareto distribution to the negative log returns with threshold 2.5%. Write down the parameter estimates and their standard errors.

(e) Obtain (i) a plot of excess distribution, (ii) a plot of the tail of the underlying distribution, (iii) a scatterplot of residuals, and (iv) a QQ plot of the residuals for the fitted GPD.

(f) Based on the fitted GPD model, compute the VaR and expected shortfall for probabilities \(q=0.99\) and 0.999 .