I have to calculate the 10-day 99% VaR of a portfolio that consists of a portfolio of 260 stocks of a company $K$ and that is short 500 call (European) options of the same company.
I know that the stocks currently have a value of €73.35, its annual volatility is 17.12% and the call options have a delta is 0.6.
I can compute the VaR of a portfolio of options but I'm a little puzzled how I can do this when we have a portfolio of a stock and an option
(I assume that by "260 stocks" you mean 260 shares of same corporation that are also the underlying of the options.)
Since the payoff of the options is non-linear, you can't get a meaningful VaR by multiplying the delta by the volatility of the underlying stock. You can't even get a meaningful VaR by including gamma or higher-order terms of a Taylor expansion.
You need to either use Monte-Carlo to generate lots of posisble market move scenarios that look like historical scenarios, or use lots of actual historical scenarios. You need to estimate the P&L of your portfolio under each scenario (there are shortcuts so you may not have to reprice under every scenario). You need to look at the worst case scenarios.
