Choosing Physical or Risk-Neutral Measures for VaR
Summary
The note explains why Value at Risk is generally estimated under the physical probability measure, which aims to describe actual future outcomes, rather than the risk-neutral measure used for pricing derivatives. Under the risk-neutral measure, expected returns are adjusted to satisfy pricing and no-arbitrage relationships; that drift is not a forecast of where an asset is likely to go. Using it for VaR can therefore distort a real-world risk estimate.
The discussion recommends estimating volatility and other distribution inputs from historical or otherwise real-world data. It notes that volatility estimates depend on choices such as the observation window, weighting method, and model. For short horizons, practitioners often set expected drift to zero to avoid blending a directional trading view into the risk estimate. This is a practical convention rather than a unique prescription: the physical measure is not uniquely specified, and the appropriate risk estimate depends on model and estimation choices. The note gives conceptual reasoning but no empirical comparison or formal derivation.
Key ideas
- VaR is intended to describe real-world loss risk, so its probability model should generally reflect physical outcomes.
- The risk-neutral measure supports arbitrage-consistent pricing and does not make its expected drift a forecast.
- Physical-measure risk estimates depend on choices such as the volatility estimator and observation period.
- Setting drift to zero can be a practical short-horizon convention that keeps trading views separate from risk measurement.
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Full text
# Which measure to determine Risk? # Which measure to determine Risk? Say I hold an equity and I want to calculate the Value-at-Risk over some period. Would one calculate the Value-at-Risk of the equity under a risk-neutral (as in martingale) measure or under the initial measure (arbitragefree and complete market assumed)? If for instance the asset price is assumed to be a Brownian motion, Risk neutrality (possibly) changes the drift and thus has significant consequences on the Value-at-Risk. So what is appropriate? And why? My guess is that one should indeed use the risk-neutral measure because the risk under the martingale measure is the ginuine risk that cannot be hedged. But I need a more throrough and formal explanation. ## Answer by Richi Wa (score 1, accepted) https://quant.stackexchange.com/a/9609 I strongly recommend not assesing risk using the risk neutral measure. Doesn't this already sound like a contradiction (risk and risk-neutral)? The risk neutral measure is there to derive prices (for derivatives e.g.) that fit to the prices of related contracts and traded assets. With "fit" I mean not allowing for arbitrage. For example if I calculate the forward price of a stock then I implicitely use the risk neutral probability measure and avoid arbitragy of trading the underlying spot. The drift/expected value is related to interest rates and dividends due to arbitrage considerations - neither due to any risk considerations nor due to any idea where the spot price of the underlying could really be in the future. If you call the other measure the physical measure then this is the one that should be used to measure risk. The problem is that it is by far not unique. For example if you measure risk by volatility (just as a starting point) then it is known that many volatility estimators exist. You can look at different observation periods, you can use weighted methods or e.g. GARCH. Very often the expected value/drift of the asset whose risk you want to measure is assumed to be zero. For short periods of time this is very reasonable. Coming up with an expected value different from zero you risk to mix up your "trading idea" with your risk measure - which is in my mind not a good thing to do.
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