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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

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  • $\begingroup$ The delta equivalent position in the stock is 260+0.6*(-500) = -40 shares of stock. Now find the VaR of this position. $\endgroup$ – noob2 Jun 29 '20 at 19:00
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(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.

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