Estimating Single-Asset VaR from Delta and Volatility

Summary

The document explains how to estimate one-day Value at Risk for a single-asset portfolio when its delta, the asset value, and daily volatility are given. It interprets delta as the portfolio’s sensitivity to a small percentage move in the underlying, then scales that sensitivity by the asset’s price and daily volatility to estimate the portfolio’s standard deviation. A normal quantile for the chosen confidence level converts that estimate into a tail-loss measure.

The worked response assumes normally distributed returns and uses the stated 98% confidence level, treating the lower-tail probability as 2%. It gives a VaR estimate with a negative sign to represent a loss. This is a delta-based approximation: it does not account for nonlinear exposure, changing volatility, non-normal returns, or other portfolio positions, so the result depends on the simplifying assumptions and the interpretation of delta.

Key ideas

• Delta translates a small percentage change in the underlying into an approximate portfolio-value change.
• Scale the portfolio sensitivity by daily asset volatility to estimate portfolio standard deviation.
• Multiply the estimated standard deviation by the normal quantile for the VaR tail probability.
• The example assumes normally distributed returns and uses the 2% tail for 98% confidence.
• A delta-based estimate omits nonlinear effects and can be inaccurate when those effects are material.

Tags

# Value at Risk from Delta of a single asset portfolio


# Value at Risk from Delta of a single asset portfolio












I am trying to figure out the following, for me unfamiliar type of question:

Given is a single asset portfolio: the Delta of the portfolio is 15, the value of the asset is 10 and the daily volatility is 2.2%. From this, I have to calculate the one-day 98% VaR of the portfolio.

I have not encountered a situation where the Delta is directly related to the VaR so I am not sure how I should approach this problem. Help is very much appreciated.

## Answer by steinbitur (score 3, accepted)

https://quant.stackexchange.com/a/10786

Given that by delta means that if the price goes up by 0.01% i.e. one basis point, you gain 15 and vice versa if the price goes down by one basis point. You know that the daily standard deviation is 2.2%, than again you know that $ 220*15 = 3300$ is the standard deviation of your portfolio. So, since we are using a normal distribution you can look at a table which describes the standard deviate for $\alpha=100\%-98\%=2\%$. If I remember correctly the standard deviate for $\alpha=2\%$ is $2.05$ and that gives you the loss measure called $VAR=-3300*2.05=-6765$

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