 # Long call option VaR with delta and gamma

Hello everyone,

Currently Im struggling with a question regarding the VaR of a long call option. I know how to calculate a 1 month VaR. The one thing I do not get is how it is affected by delta and gamma.

Kind regards,
Luc

This is what i did. i didnt use gamma, and instead relied on an approximation formula.

A long call option with a delta of 1.50 can be tought of as a portfolio of 1.50 shares (a delta of 1.50 means that the option has the risk of 1.50 shares)
there is an approximate formula, assuming stocks are lognormally distributed and h is small enough so that the distribution is approximately normal (from Derivatives Markets by Robert McDonald)

Sh = S0 * (1 + alpha*h + Z * sigma * sqrt(h) )
if h is sufficiently small, the formula simplifies further to
Sh = S0 * (1 + Z * sigma * sqrt(h) )
that is, you can ignore the effect of return when h is small enough

where S0 is the initial stock price, alpha is the expected rate of return on the stock, sigma is the annualized volatility, and h is the investment horizon.
for this problem,
S0 = 14.13 * 1.50, Z = 2.326, sigma = .03 * sqrt(12) and h = sqrt(1/12)
using the second formula, which ignores the expected return, we have
Sh = 14.13 * 1.50 * ( 1 + 2.326 * .03) = 22.6739871
VaR = Sh - S0 = 22.6739871 - 14.13 * 1.50 = 1.4789871
or approx. 1.479, which matches the answer provided.