Setting Expected Returns to Zero in Short-Horizon VaR
Summary
The document considers whether stock drift can be approximated as zero when calculating value at risk over a short horizon. In a geometric Brownian motion setup, expected return accumulates in proportion to time while volatility scales with the square root of time. For horizons of one to ten days, the response says the drift contribution is generally small relative to volatility, making a zero-drift approximation reasonable for stocks in that setting.
It cautions against combining zero expected return for stocks with a risk-free-rate drift assumption for options without considering the resulting inconsistency in the assumed risk premium. Historical average returns are mentioned as one possible estimate of expected return, but the response questions their general suitability because observed prices reflect physical probabilities while option pricing commonly uses risk-neutral probabilities. These are modeling guidelines, not a universal VaR rule; the approximation depends on the model, horizon, and purpose of the calculation.
Key ideas
- In a geometric Brownian motion model, drift scales with time while volatility scales with the square root of time.
- For short horizons, setting stock expected return to zero can be a reasonable VaR approximation.
- Using different drift assumptions for stocks and options can create an inconsistent risk-premium setup.
- Historical average returns may not provide a suitable expected-return estimate for option-pricing contexts.
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Full text
# Is it OK to consider the expected return is zero for stocks when calculating VaR over a short horizon?
# Is it OK to consider the expected return is zero for stocks when calculating VaR over a short horizon?
I want to implement the approach described in the following recipe for calculating VaR: Is there a step-by-step guide for calculating portfolio VaR using monte carlo simulations
I was told that I can safely assume that $\mu=0$ for stocks when calculating VaR over a short time horizon (1 to 10 days). Is this correct? The reasoning would be that $\sigma$ will be much larger than $\mu$, which will be negligible in comparison (and close to zero). In that case can I simply use $\mu=0$ for stocks and $\mu=r$ for options in the step-by-step approach described in the previous link? Also how should I approximate $\mu$ in the general case if I needed to? Should I use historical stock returns over a period of time (for example 100 days), average it and consider the result as an approximation of $\mu$? Thanks for your help!
## Answer by user32416 (score 2, accepted)
https://quant.stackexchange.com/a/20809
- Yes, for a short time horizon like 1 - 10 days, assuming $\mu = 0$ is fine. As you'd correctly pointed out, for 1 - 10 days (and referring to the link you'd referenced to), it scales linearly by $T$ (recall that $T$ is an annual number, so convert to a % number in reference to days), but volatility scales by $\sqrt{T}$ and so it is much larger than $T$ for small $T$'s like the time horizon you're considering. This is especially true since you're using a GBM type setup.
- It seems to me that you'd be somewhat inconsistent if you use $\mu = 0$ for stocks and yet $\mu = r$ for options, since then your risk premia would be $\mu - r = 0$.
- I suppose that that is one approach to estimate $\mu$. However, given that people usually work with risk neutral probabilities when pricing options, and the actual prices you observe are physical probabilities, it is unclear whether that is a good estimate of $\mu$ in general.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.