# Convert drift and diffusion term in terms of time in the Geometric Brownian Motion framework

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.

• 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 ? Mar 1, 2020 at 21:59
• @markleeds Hi . Just approximating. I could also use 250. Mar 2, 2020 at 7:58
• 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. Mar 2, 2020 at 10:18
• 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. Mar 2, 2020 at 11:45
• I'm pretty sure that's correct and glad we're on the same page. Mar 2, 2020 at 18:12