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Assume that we have daily prices covering the period of 10 years. For calibrating the drift and diffusion parameters of the GBM model $$S_{t+1} = S_{t}e^{[(\mu-\sigma^2/2)]\Delta t + \sigma \sqrt{\Delta t} Z_{t}}$$ I use the following formulas: $$\sigma = \sqrt{\frac{Var[R]}{\Delta t}}$$ $$\mu = \frac{E[R]}{\Delta t} + \frac{\sigma^2}{2}$$

Where R are the daily return series formed by the daily prices that I pre-mentioned. Since I have 10 years of daily observations I use $$\Delta t= \frac{10}{260}$$

The drift and diffusion results that I get from above are let's say "daily" since they are based on daily returns. My question is how we can convert them to weekly? I know that for the volatility we simply need to multiply by square root of 5 i.e. $$\sigma_{w}=\sigma \times \sqrt{5}$$
but for the drift I do not know how to convert it.

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  • $\begingroup$ Hi: do you really have EXACTLY 260 days per year or are you approximating ? I would think, with holidays etc, you would lose some data ? $\endgroup$
    – mark leeds
    Commented Mar 1, 2020 at 21:59
  • $\begingroup$ @markleeds Hi . Just approximating. I could also use 250. $\endgroup$
    – Whitebeard13
    Commented Mar 2, 2020 at 7:58
  • $\begingroup$ So, it sounds like an approximation Given, that you're willing to deal with the approximaton, then. for the mean, $\mu$ do the same type of scaling. So, use the same formula for $\mu$ but multiply the first term by 5 and, for the second term, use the 5 day volatility. I think that's correct but let me know if that makes sense to you. $\endgroup$
    – mark leeds
    Commented Mar 2, 2020 at 10:18
  • $\begingroup$ Hi. Thanks for your answer. It's quite funny. I thought the same. I was actually between two options. Either multiply by 5 the first term plus incorporate the $\sigma_{w}$ or multiply by 5 the whole "daily" $\mu$. I ended up using the second as the assignment was requesting to use the "daily" $\mu$ for deriving the "weekly" one. $\endgroup$
    – Whitebeard13
    Commented Mar 2, 2020 at 11:45
  • $\begingroup$ I'm pretty sure that's correct and glad we're on the same page. $\endgroup$
    – mark leeds
    Commented Mar 2, 2020 at 18:12

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