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Scaling GBM Drift and Volatility Across Time Intervals

Article Quant Q&A · Author: jessica
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Summary

The document explains how to relate daily estimates of drift and volatility to the annual parameters commonly used in geometric Brownian motion. It also corrects a notation error: a Brownian motion increment is random, so it cannot be equated to a deterministic volatility factor times the square root of time. Brownian motion has zero expected value and variance that grows with elapsed time.

Using a convention of 252 trading days per year, the answer gives the standard annualization rules: multiply daily drift by 252 and daily volatility by the square root of 252. This lets daily return estimates be used with a model parameterized on an annual basis. The guidance assumes the stated trading-day convention and the usual scaling framework; the document does not examine how estimation noise, serial dependence, or other departures from model assumptions might affect these conversions.

Key ideas

  • Brownian motion increments are random, not deterministic quantities proportional to the square root of time.
  • Brownian motion has zero expected value and variance proportional to elapsed time.
  • Under a 252-trading-day convention, annual drift is daily drift multiplied by 252.
  • Annual volatility is daily volatility multiplied by the square root of 252.
  • The scaling rules rely on the usual GBM framework and the chosen trading-day convention.

Tags

StatisticsVolatilityEquities
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# Scaling Intervals in Diffusion Process


# Scaling Intervals in Diffusion Process












I know this is a very elementary question but... when modeling asset prices through a stochastic process as in

$$dS_t=S_t μ dt+S_t σdW_t,$$

where the following is a wiener process $$dW_t=σN(0,1)dt^{1/2}$$

how does the mean $μ$ and volatility $σ$ scale with the time interval $dt$?

If I am forecasting for 1 year in the future, in 1 day steps, $dt$=1/250=.004

But what if the mean parameter I have calculated for $μ$ and $σ$ is for daily returns, would the equation still hold? ie, I take a 20SMA of returns for the last 20 days, so my return is already daily. In all the literature I have read $μ$ and $σ$ are already annual which in my case is not helpful because I look at daily returns for assets not annual returns that have been scaled down to daily.

## Answer by HardyHulley (score 2)

https://quant.stackexchange.com/a/9706

First, your statement that $dW_t=\sigma\,dt^{1/2}$ is incorrect. In fact, it's not even meaningful (you can see this by noticing that the expression on the left-hand-side is an "increment" of Brownian motion, and hence random, while the expression on the right-hand side is deterministic). What you mean to say is that $W$ is a Brownian motion, and hence $\text{E}(W_t)=0$ and $\text{Var}(W_t)=t$. Or better, the quadratic variation of $W$ is $<W>_t=t$. It's important to remember that the "increment" $dW_t$ is simply a notational convenience - it's not really a well-defined concept, since the paths of Brownian motion have infinite first variation.

Now, coming to your question, the in asset pricing models the parameters $\mu$ and $\sigma$ are usually specified as the annual drift rate and the annual volatility. If instead, you have estimated a daily drift rate $\mu_d$ and a daily volatility $\sigma_d$, then you can scale them as follows $\mu=252\mu_d$ and $\sigma=\sqrt{252}\sigma_d$, where we're using the convention that a year contains 252 trading days.

Regards Hardy

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