$3.$ Consider a portfolio of options on four stocks, with share price,

portfolio delta and annual volatility of each stock as follows $($see pictrue$)$: The correlations between the stocks are as follows $($see picture$)($a$-$b$)$ Use the Linear Model to estimate the $20-$day $98\%$ Expected Shortfall $($ES$)$ for each of the four stocks

separately using the delta position of each stock in the table above $($i$.$e$.$ calculate four separate ES results$).$

Which two stocks have the largest and smallest ES and how do each stock$$s position and volatility affect

that order?

$($c$-$d$)$ Use the Linear Model to estimate the $20-$day $98\%$ Expected Shortfall for the portfolio as a whole $($i$.$e$.$

calculate one single ES result$).$ What is the benefit of diversification compared to part $($a$-$b$)?$ What is the

marginal ES of FESTI $($i$.$e$.$ by how much does the ES change if the position in FESTI is removed from the

portfolio$)?$

$($e$)$ How and why do the results change if you use the $20-$day $98\%$ Value at Risk $($VaR$)?$ks are as follows $($see picture$).$