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Estimating Drift and Volatility for Geometric Brownian Motion

Article Quant Q&A · Author: Lopo
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Summary

The discussion addresses how to interpret and estimate the parameters used to simulate asset prices with geometric Brownian motion. It corrects the stated solution formula: under the usual GBM convention, the log-price drift includes a minus half variance adjustment, and Brownian motion is scaled by volatility. Consequently, the mean of log returns is not itself the price-process drift parameter.

For equally spaced log returns, the reply describes estimating volatility with their sample standard deviation, then estimating drift by adding half the estimated variance to their sample mean. Parameters must be expressed consistently with the time step; the second answer illustrates the idea using annualized return and volatility figures. The discussion cautions that GBM is a simple model and may fit stock behavior poorly. It does not prescribe a lookback window or settle whether rolling or realized volatility is best for a particular application.

Key ideas

  • The GBM log-price solution includes a half-variance adjustment to the drift and scales Brownian motion by volatility.
  • The sample mean of log returns estimates log-return drift, not the GBM price-process drift directly.
  • Volatility can be estimated from the sample standard deviation of log returns.
  • The drift estimate adds half the estimated variance to the mean log return.
  • Parameter estimates must use units consistent with the simulation time step, and GBM may be too simple for stocks.

Tags

EquitiesStatisticsVolatilityBacktesting
Full text
# About SDE of Geometric Brownian Motion


# About SDE of Geometric Brownian Motion












It's known that most of the financial assets are subject to Geometric Brownian Motion, which satisfies the following equations:

$\frac{dS}{S}=\mu dt + \sigma dX$ (1)

$S_t = S_0 e^{(\mu + \frac{1}{2} \sigma^2)t + X_t}$ (2)

Here my questions are:

In practice, when we use this SDE to simulate asset price path, i.e. the price movement of a stock index, what parameters should I use for $\mu$ and $\sigma$.

Does the $\mu$ stand for annualized mean log return of the stock index? But in equation(1), the $\mu$ doesn't appear as the exponent of e. Although, in equation(2) $\mu$ is really the exponent of e.

Does the $\sigma$ stand for the volatility of stock price, or volatility of logarithm of price, or volatility of log returns?

Here the volatility is an annualized or not?

The volatility is a rolling volatility or realized volatility?

thanks for your attention!

## Answer by userid is i (score 1)

https://quant.stackexchange.com/a/40690

Equation (2) is wrong, it is is $(\mu - \sigma^2/2)t$ and $\sigma X_t$. Then estimating $\mu$ is not just taking the mean of log returns. Using $\log(S_{n+1}) - \log(S_{n}) \sim N(\mu - \sigma^2/2, \sigma^2)$, you estimate $\sigma$ with the sample standard deviation of log returns, say $\hat \sigma$, and then you get an estimate for $\mu$ with "sample mean of log returns+ ${\hat \sigma}^2/2$." And actually GBM is not a very good model for most stocks, it is too simple.

## Answer by eSurfsnake (score 0)

https://quant.stackexchange.com/a/39203

It's all in return units. Assume, for example, that from 1950 to 2017 annualized average returns on the S&P 500 are about 11.5% (.115) and standard deviation/vol is about 17% (0.17). That is you $\mu$ and $\sigma$.

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