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Estimating Time-Varying GBM Drift and Volatility from Long Histories

Article Quant Q&A · Author: randomWalk
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Summary

The document raises the problem of estimating drift and volatility for Geometric Brownian Motion (GBM) from a 50-year financial time series. It notes that fitting one pair of parameters to the entire history may be a poor approximation because expected returns and volatility can change over time, and because GBM may be less suitable across long horizons.

One proposed approach is to let drift and volatility vary deterministically with time, estimating values year by year and updating the GBM parameters. The document asks whether this approach is sensible but provides no estimation procedure, data analysis, simulation results, or resolution. Its main lesson is to recognize the assumptions behind full-sample parameter estimates and consider nonstationarity when using GBM for long histories. Whether annual updates improve forecasts or simulations depends on the data and model assumptions, which the document does not assess.

Key ideas

  • A single full-history estimate assumes drift and volatility are stable over the sample.
  • The document notes that asset returns and volatility can change over time.
  • It proposes considering time-dependent GBM parameters estimated in yearly intervals.
  • It does not provide evidence that this approach improves model fit or forecasts.

Tags

StatisticsVolatilityBacktesting
Full text
# how to estimate Geometric Brownian Motion parameters on long timeseries


# how to estimate Geometric Brownian Motion parameters on long timeseries












I'm working on a 50-years financial timeseries and I would like to simulate GBM paths from it. The first thing I'm supposed to do is to estimate the drift $\mu$ and the volatility $\sigma$ parameters.

What I've done so far is to come up with the estimate on the entire history, but I think it's quite a big approximation since the expected return and the volatility of a financial asset vary over time. Also, it is known that GBM doesn't work so well on long time interval.

I thought that I could have time-dependent deterministic parameters for $\mu = \mu(t)$ and $\sigma = \sigma(t)$ where I compute the values for every years and update the parameters of the GBM accordingly, but I'm not sure if it makes sense.

Any suggestion?

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