Consider a delta-neutral position in a single asset with a gamma (measured with respect to percentage changes in the asset) of g (g > 0). Suppose that the 10-day return on the asset is normally distributed with a mean of zero and a standard deviation s. 

(a) What is an expression for the change in the portfolio value, DP, over 10 days as a function of g, s, and a random sample from a standard normal distribution? 

(b) The square of a standard normal is chi-squared with one degree of freedom. Write the expression for DP in (a) as a function of a random sample from such a chi-squared distribution and C where C = gs²/2.

(c) The 99th percentile of a chi-squared distribution with one degree of freedom is 6.63. (See CHISQ.INV in Excel.) Use this to show that the 99th percentile of DP is 6.63C. 

(d) The mean and variance of a chi-squared distribution with one degree of freedom are 1 and 2. Show that this result is consistent with Figure 17.3 and the formula for IM (Gamma) in Section 17.2.2. 

(e) Show that when g < 0 the 99th percentile of DP is less than zero and that Figure 17.3 then correctly gives IM (Gamma) = 0.