Using Volatility Inputs in Normal-Distribution Parametric VaR

Summary

The document asks whether an implied volatility index can supply the volatility input for a one-day parametric Value at Risk estimate on a stock index. It notes that the volatility index reflects an expected horizon of about a month, while the VaR horizon is one day, and raises the issue of rescaling annualized volatility to a daily figure. The reply instead outlines the basic parametric VaR setup: estimate a distribution from historical daily returns, assume normality, and determine the distribution using its mean and standard deviation. Once those parameters are specified, a quantile-based VaR can be calculated analytically.

The response explains this general framework but does not directly evaluate using implied volatility as the forecast input, resolve the horizon mismatch, or discuss other potential problems. Its practical guidance is therefore limited: it clarifies what a normal parametric VaR requires, while leaving the central comparison between implied and historical volatility open. Normality is also an assumption, not evidence that return tails or changing volatility are adequately captured.

Key ideas

• Normal parametric VaR models daily returns using an assumed probability distribution.
• Under a normal assumption, the mean and standard deviation determine the return distribution.
• A specified quantile of that distribution gives an analytical VaR estimate.
• The reply does not settle whether a one-month implied volatility measure is appropriate for a one-day risk horizon.

Tags

# Implied volatility in parametric VaR


# Implied volatility in parametric VaR












I'm calculating 1-day parametric VaR estimates for a stock index under the simple assumption that the returns are normally distributed. My question is, what is your opinion of using a volatility index such as the VIX as an input for the expected volatility of the underlying stock index?

The (annualized) volatility index level would be re-scaled at the daily level (e.g. ${\sqrt{1/360}}\,VIX$). I recognize that VIX is an expectation over one month which is longer than the one-day VaR risk horizon. Are there some other fundamental problems?

## Answer by JejeBelfort (score -1)

https://quant.stackexchange.com/a/33602

The idea of the parametric VaR is to fit a parametric distribution to historical data. In your case, the historical data is your daily returns.

Starting from this, you have to make an assumption on this historical returns distribution. The most obvious choice is to assume a normal distribution, which is entirely determined by the mean (average of your historical returns distribution) and the variance (standard deviation of your daily historical returns distribution).

Once you have fully parameterized your distribution, there exists a closed-form solution for the VaR at any quantile $\alpha$. Here is a post mentioning this formula:

Parametric/Analytical VaR

Hope this helps

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