Computing VaR with Gamma-Distributed Volatility

Summary

The document asks how to calculate a portfolio’s 99% value at risk when returns are normal conditional on volatility, but volatility itself follows a gamma distribution. It contrasts this setup with the familiar normal VaR calculation based on fixed volatility and points toward a Bayesian mixture approach.

The response suggests using conjugate priors for a normal distribution with known mean, under which integrating over uncertain dispersion produces a Student-t return distribution with an updated dispersion parameter. VaR can then be obtained from the corresponding tail quantile. The document does not derive the parameter update, give a numerical procedure, or report a worked example, so the exact result depends on specifying the distribution parameterization and the relationship between gamma volatility and normal variance. Those details need verification before applying the suggestion to a portfolio.

Key ideas

• Uncertain volatility can be modeled by mixing conditional normal return distributions over a volatility distribution.
• The response links this mixture to conjugate-prior results for a normal model with known mean.
• It suggests a Student-t predictive return distribution and using its tail quantile for VaR.
• The document gives no derivation or example, and the precise parameter mapping is left unspecified.

Tags

# Value at Risk (VaR): Normal distribution with gamma distributed volatility


# Value at Risk (VaR): Normal distribution with gamma distributed volatility












If I was to do a 99% VaR calculation on a portfolio with normally distributed returns $\mathcal{N} (\mu,\sigma)$, the 99% VaR would be $\mu - 2.33\sigma$.

Instead of having a constant volatility, let's say volatility is gamma distributed, i.e. $\sigma \sim \Gamma(k, \theta)$.

Explain in as many details as possible (either derive a formula or explain a numerical solution using a computer program) how to compute the VaR of the portfolio when returns have a normal distribution conditional on sigma, and sigma is distributed according to a gamma distribution.

## Answer by Kermittfrog (score 1)

https://quant.stackexchange.com/a/53076

You might want to have a look at the conjugate priors to the normal distribution (with known mean) Your setup will result in a $t$-distributed return with updated dispersion parameter.

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