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1

The "right" thing to do is to treat the options as derivative contracts. Let's say for simplicity that you are using Monte Carlo to compute VaR. Then you would simulate the equity prices on each iteration, and then apply an option-pricing formula to get the corresponding option prices on that iteration. This lets you obtain an accurate simulated portfolio ...

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The Black-Scholes delta: $$\partial_SC=N\left(\dfrac{\ln\left(\frac{S_0}{K}\right) +(r - q + \frac{1}{2}\sigma^2)(T - t)}{\sigma\sqrt{T - t}}\right)$$ As you can see this delta would go to$1$ if $\sigma\to\infty$ (and $t<T$).

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consider adjoint algorithmic differentiation to get an exact derivative here. Works especially well for monte carlo. Here is an example paper: http://luca-capriotti.net/pdfs/Finance/jcf_capriotti_press_web.pdf

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