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For an option with price C, the ΔC, with respect to changes of the underlying asset price S and volatility σ (first-order approximation), is given by

$\Delta C=\delta \Delta S+\nu\Delta\sigma$,

where δ, and ν are respectively the delta, and vega greeks.

Assuming the asset S and the volatility σ as normal and indipendent, we can calculate the percentual VaR of the option by using a parametric VaR as follows:

$VaR = −\alpha*\sigma_p$,

where $\sigma_p^2$ is the portfolio variance: $\delta^2*\sigma_2^2 + \nu^2*\sigma^2$,

where σ_s is the underlying S volatility, and σ_sigma is the volatility of implied volatility.

How to derive the numeric VaR (in terms of money) ? By multiplying the percentual VaR by the position my portfolio ? What is the latter ? Is it δS + νσ ? But I already included delta and vega in the portfolio volatility calculation ?

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If you want to calculate the option VaR using historical data, i think the first ste is to take your historical spots, volas, rates etc, then calculate. the changes in the spot prices, take the percentile of each time series of risk factors (e.g. in Excel use PERCENTILE.INC() ) and multiply these with your greeks (i.,e. delta*(PERCENTILE.INC() etc.). You do this with your changes in rates and implied vola, multiply these with your greeks . Keep in mind you are using a Taylor Approx. Good luck

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  • $\begingroup$ Thanks. But doing this way, you are summing up the each single VaRs. Is this correct ? VaR is not additive. The VaR of a portfolio is not sum of the underlying VaRs. $\endgroup$
    – userLx
    Feb 21, 2023 at 14:09
  • $\begingroup$ This is correct. I wrote a bit cryptic that this is a Taylor approximation, not the sum of VaR. If you want to calculate the VaR of a portfolio of options, by summing the single VaR component you will have this subadditivity. But that's another story.. $\endgroup$
    – T123
    Feb 21, 2023 at 17:25

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