Question

On Black Monday, the return on the S&P 500 index was 22.8%. Ouch! This excercise attempts to answer the question, "what was the conditional probability of a return this small or smaller on Black Monday?" "Conditional" means given the information available the previous trading day.

[a] Download daily returns on the S&P 500 index from the R package Ecdat using the following command data(SP500, package='Ecdat') You need to install the package first if your computer does not have it installed before. The S&P 500 returns are in the dataset SP500. The returns are the variable r500, which spans the period from January 1981 to April 1991.

[b]Black Monday is the 1805th return in this dataset. Fit a GARCH(1,1) model to the last two years of data before Black Monday, assuming 250 trading days per year. What are the estimates of the parameters of the model?

[c] Make a plot of the fitted volatility during the chosen two year period.

[d] Calculate the conditional probability of a return less than or equal to 0.228 on Black Monday (Hint: the mean of the return can be taken to be zero).

[e] Compute and plot the standardized returns. Also plot the autocorrelations of the standardized returns and their squares. Do the standardized returns indicate that the GARCH(1,1) model fits adequately?

[f] Are the standardized returns normally distributed? If not, does it have fatter or thinner tails than a normal distribution? Justify your answers.

[g] Fit a GARCH(1,1) model with a t-distributed shock using the following command fit = garchFit( garch(1,1), data=SP500$r500, cond.dist='std') The extra argument cond.dist='std' in the garchFit function indicates the shock in the GARCH model is t-distributed. Among the parameter estimates, the new param- eter shape gives the estimated degree of freedom of the t distribution. Recalculate the conditional probability of a return less than or equal to 0.228 on Black Monday. Also you can use the pt(q, df) function to get the probability of a t distribution given the quantile q and the degree of freedom df.

Please explain with R coding!!!!!!!