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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!

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  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.

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  • $\begingroup$ 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? $\endgroup$ – BigONotation Sep 20 '15 at 0:03
  • $\begingroup$ 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? $\endgroup$ – BigONotation Sep 20 '15 at 0:11
  • $\begingroup$ 1) Seems right to me. 2) You need to work on your notations, namely that don't use the same notation to denote two different things. I think you're talking about the drift of asset price process after measure change in the Black Scholes model. And yes, after applying Girsanov's theorem, the drift of the risky asset price process is that of the risk free rate. But under the physical measure, the risky price process drift is still whatever it was originally. $\endgroup$ – user32416 Sep 20 '15 at 1:29
  • $\begingroup$ Yes indeed I need to pay attention to my notation. Thanks again for the clarifications! $\endgroup$ – BigONotation Sep 20 '15 at 10:12

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