This problem is based on the data contained in the script file SPfutures.asc which creates a matrix SPFUT with two columns, each row corresponding to a given day. The first column gives, for each day, the log return of a futures contract which matures three weeks later, (we'll call that variable X), and the second column gives, on the same day, the log return of a futures contract which matures one week later (we'll call that variable Y ).

Question 2 is not required for the rest of the problem. In other words, you can answer questions 3 and 4 even if you did not get question 2.

1. Compute the means and the standard deviations of X and Y , and compute their correlation coefficients

2. We first assume that X and Y are samples from a jointly Gaussian distribution with parameters computed in part 1. For each value α = 25%, α = 50% and α = 75% of the parameter α, compute the q-percentile with q = 2% of the variable αX + (1 - α)Y

3. Fit a generalized Pareto distribution (GPD) to X and Y separately, and fit a copula of the Gumbel family to the empirical copula of the data.

4. Generate a sample of size N (where N is the number of rows of the data matrix) from the joint distribution estimated in question 3.

4.1. Use this sample to compute the same statistics as in question 1 (i.e. means and standard deviations of the columns, as well as their correlation coefficients) and compare to the numerical values obtained in question 1.

4.2. Compute, still for this simulated sample, the three percentiles considered in question 2, and compare the results.