Limits of Brownian Motion Models for Financial Prices

Summary

The document outlines three limitations of Brownian motion as a model for financial prices. Its continuous paths cannot represent sudden price jumps, and its Gaussian assumptions may fail to capture the distribution of market returns. It also raises stationarity as a major concern because financial time series can change over time, making fixed statistical properties an unrealistic assumption. These points caution against treating a Brownian model as a complete description of market behavior.

The discussion notes that the model can still be useful for long-run simulation. It also cites evidence that daily S&P 500 log-price changes showed small but statistically significant correlations at short lags over a multi-decade sample, contrary to the uncorrelated increments implied by geometric Brownian motion. The material is brief: it does not quantify the effects of these limitations, compare alternative models, or specify when the long-run simulation claim holds. A further comment characterizes the model as a binomial tree in another form, without explaining that equivalence.

Key ideas

• Brownian motion assumes continuous price paths and therefore omits jumps.
• Gaussian return assumptions can miss features of observed financial returns.
• Financial time series may not be stationary, limiting the use of fixed model parameters.
• The cited S&P 500 evidence reports small but significant short-lag correlations in daily log changes.
• The document suggests Brownian motion can remain useful for some long-run simulations.

Tags

# What are the limitations of brownian motion in finance?


# What are the limitations of brownian motion in finance?












What are the limitations of brownian motion in its applications to finance?

## Answer by ast4 (score 5)

https://quant.stackexchange.com/a/899

So where to begin? Continuity is a big thing as it fails to take into account jumps, the Gaussian assumption is another big one. However, looking deeper into it stationarity is a huge problem as it applies to financial time series.

However, it does an OK job at simulation stuff in the long-run.

## Answer by chrisaycock (score 4)

https://quant.stackexchange.com/a/898

From this paper:

> The geometric Brownian motion model implies that the series of first differences of the log prices must be uncorrelated. But for the S&P 500 as a whole, observed over several decades, daily from 1 July 1962 to 29 Dec 1995, there are in fact small but statistically significant correlations in the differences of the logs at short time lags.

## Answer by Keith A. Lewis (score 4)

https://quant.stackexchange.com/a/900

It is a binomial tree in disguise.

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