Suppose that the daily log returns of an asset follow a location-scale \(\mathrm{t}\) distribution with mean 0 , scale 0.002 , and degrees of freedom 6, and assume that the daily log returns are sequentially independent. Suppose further that the prices of the asset are subject to random shocks that follow a Poisson process. Assume the perturbations occur on average 4 times per year.

(a) What value of the Poisson parameter will yield a mean of 4 events per year (253 trading days)?

(b) Assume that each shock results in a \(4 \%\) positive jump in the price of the stock with probability 0.4 , and a \(7 \%\) negative jump in the price of the stock with probability 0.6 .
Generate and plot (on the same graph) 5 random sequences of 253 daily prices of the asset, each starting at \(\$ 100\).