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Estimating Option VaR with Delta and Vega Sensitivities

Article Quant Q&A · Author: userLx
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Summary

The document outlines a parametric approach to option value-at-risk using a first-order Taylor approximation. It estimates option price changes as the sum of delta exposure to underlying-price changes and vega exposure to implied-volatility changes. Under the stated assumption that the underlying and volatility changes are independent and normally distributed, their Greek-weighted variances can be combined to estimate the option’s return volatility and a percentile VaR.

The question asks how to convert that percentage risk estimate into a monetary amount and what portfolio value should be used. The answer instead suggests a historical simulation approach: collect past changes in relevant risk factors, select a percentile for each, and multiply those changes by the corresponding Greeks. This remains a linear approximation, so it may miss curvature and interactions; the response does not resolve the monetary scaling question or specify a complete position-sizing convention. The method also relies on the independence and normality assumptions for its parametric form.

Key ideas

  • A first-order option price change can be approximated using delta and vega exposures.
  • Independent normal changes in spot and implied volatility allow their Greek-weighted variances to be combined.
  • A parametric VaR can be derived from the resulting estimated price-change volatility and a chosen percentile.
  • Historical risk-factor percentiles can also be multiplied by the corresponding Greeks for a scenario estimate.
  • Both approaches use a Taylor approximation and may omit nonlinear option effects.

Tags

OptionsRisk managementStatistics
Full text
# How to derive numeric option VaR with delta-vega normal approach?


# How to derive numeric option VaR with delta-vega normal approach?












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 ?

## Answer by T123 (score 0)

https://quant.stackexchange.com/a/74713

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.