Delta-Normal and Delta-Gamma Value-at-Risk Approximations

Summary

The note compares two approximations for portfolio Value-at-Risk when market risk factors are modeled as Gaussian. Delta-Normal treats portfolio value as a linear function of those factors. The resulting portfolio return remains Gaussian, with variance calculated from the portfolio sensitivities and the factors’ covariance matrix, so VaR follows directly from a Gaussian quantile.

Delta-Gamma adds a quadratic term to represent curvature in portfolio value. This can improve the approximation for nonlinear exposures, but the resulting portfolio distribution is generally no longer Gaussian, even when the underlying factors are. The note says its distribution must instead be evaluated with a semi-explicit numerical technique, such as applying a fast Fourier transform to the moment-generating function. It offers no numerical comparison or empirical validation, and does not detail assumptions or implementation choices. The practical tradeoff is a potentially more realistic treatment of curvature at the cost of a less tractable VaR calculation.

Key ideas

• Delta-Normal approximates portfolio value as a linear function of Gaussian risk factors.
• The linear approximation preserves a Gaussian portfolio distribution, allowing VaR to be obtained from its variance and a Gaussian quantile.
• Delta-Gamma includes a quadratic sensitivity term to capture portfolio curvature.
• The quadratic term generally makes portfolio returns non-Gaussian, even when risk factors are Gaussian.
• Delta-Gamma VaR can require numerical distribution methods such as Fourier transforms.

Tags

# Difference between Delta-Gamma and Delta-normal method for VaR


# Difference between Delta-Gamma and Delta-normal method for VaR












I was reading the differences between Delta-Gamma and Delta-normal method for VaR. One of the difference I found is mentioned below, but I can't understand it's importance. Can anybody please explain this?

> The improved accuracy by the Delta-Gamma method comes at the cost of at least some reduced tractability relative to the Delta-Normal model. In using Delta- Gamma approach we might lose normality in our portfolio return even if changes in the underlying risk factors are normally distributed

Thanks

## Answer by Antoine Conze (score 2)

https://quant.stackexchange.com/a/39142

Delta-Normal VaR means the portfolio is approximated as a linear function $a + \Delta^T X$ of gaussian risk factors $X$ with variance covariance $V$, so the portfolio distribution is also Gaussian, with variance $\Delta^T V \Delta$, and its VaR is computed explicitly as a percentile on the Gaussian probability distribution function.

Delta-Gamma VaR means the portfolio is approximated as a quadratic function $a + \Delta^T X + \frac{1}{2} X^T \Gamma X$ of gaussian risk factors $X$. The portfolio distribution is no longer Gaussian (because of the quadratic term) and its probability distribution function must be computed trough a semi-explicit numerical method such as a fast Fourier Transform applied to its moment generating function.

See Stefan R. Jaschke risk management for financial institutions http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.543.8751&rep=rep1&type=pdf for a good overview of this topic.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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