# 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!

1. 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.
2. 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 $\mu - r = 0$.
3. 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.
• 1) @user32416 Thanks for your clarifications! For a 1-day VaR, I would use $T=1/252$ assuming that their are 252 days of trading. Correct? – BigONotation Sep 20 '15 at 0:03
• 2) @user32416 My understanding is that if I am going to price any options in the portfolio using the Black-Scholes Framework, I must assume that the stock will grow at the risk free rate $\mu=r$, no matter what the actual rate. So in the option pricing formula I would use $r$.Am I missing something? – BigONotation Sep 20 '15 at 0:11