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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 ...

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