How do I compute Value at Risk of a European call option?

Question

Consider a European call option on a non-dividend paying stock, where the option has strike K = 100 and expiry T = 0.25, i.e. the option expires 3 months from now. The option is on a single share. The current price of the stock is 100, the riskfree interest rate is zero, and the option premium (i.e. price) is equal to the price given by the Black-Scholes formula using a volatility of 30% (where this is quoting volatility on an annualized basis). The expected return on the stock is 10% annually.

Let Π denote a portfolio consisting of a long position of 100, 000 of these options. At a confidence level of 95%, what is the 1-day Value-at-Risk of the portfolio Π?

What changes it instead of a long position I take it short?

Answer 1

Quick back-of-the-envelope calculation would be to say that the downmove which will not be exceeded in 95% of cases, is $N^{-1}(0.05)=-1.64$ of a daily return standard deviation $\sigma_{daily}=30\%/\sqrt{365}=1.57\%$. So that means a downmove of $-1.64*1.57\%=-2.58\%$. Your option is ATM, so ca. 50% delta or the equivalent of 50'000 shs. 50'000 shs at price 100 correspond to 5'000'000 invested, a $-2.58\%$ move on this is ca. -129'000. For a more accurate value, you would have to include gamma, which on your option position is long, hence VaR will turn out slightly smaller than the -129'000 done with this simple analysis.

On a short position, the analysis remains the same for delta, gamma however will be systematically to your disadvantage, making VaR<-129'000.