Annualizing Geometric Brownian Motion Drift and Volatility

Summary

The document describes calibrating a geometric Brownian motion model from daily prices. It gives formulas for estimating diffusion from the variance of daily returns and drift from their mean, adjusted by half the estimated variance. The questioner then asks how to express these parameters on a weekly time scale, noting the familiar square-root-of-five scaling for volatility.

The key modeling issue is that drift and diffusion scale differently with time: the prompt identifies volatility scaling but does not provide an answer for weekly drift. It is therefore a question about converting parameter units, not a completed derivation or validated calibration procedure. It also gives a ten-year sample and a daily time-step expression, but does not clarify the return convention or reconcile the stated observation period with that expression. Those details matter when interpreting estimates, and the document itself leaves the weekly drift conversion unresolved.

Key ideas

• The setup estimates GBM diffusion from daily return variance and drift from mean return with a variance adjustment.
• The questioner proposes scaling daily volatility by the square root of five for a weekly horizon.
• The document raises but does not answer how drift should be converted from daily to weekly units.
• The daily time-step expression and return conventions are not further examined.

Tags

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


# 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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.