Why Brownian Motion Is Used to Model Financial Prices
Summary
The document explains why Brownian motion is a useful starting point for modeling financial markets. It presents Brownian motion as the continuous-time limit of a scaled random walk and as a simple, well-studied process with rough, irregular paths that resemble observed price movements. Its Markov and martingale properties also make it mathematically convenient for financial models and stochastic calculus.
The discussion connects Brownian motion to option pricing: assuming log returns follow Brownian motion with drift leads to closed-form option prices. It also sketches the historical development from Bachelier’s early price model to Samuelson’s use of geometric Brownian motion for positive stock prices and the later Black–Scholes formula. These are motivations and modeling examples, not evidence that real prices follow Brownian motion. The document does not examine empirical fit or limitations such as jumps, changing volatility, or market frictions.
Key ideas
- Brownian motion can be viewed as a continuous-time limit of a scaled random walk.
- Its irregular paths offer a simple model for fluctuating financial prices.
- Markov and martingale properties make Brownian motion useful in financial mathematics.
- Brownian log returns with drift enable tractable option-pricing models.
- Geometric Brownian motion was introduced to keep modeled stock prices positive.
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# Why is Brownian motion useful in finance? # Why is Brownian motion useful in finance? The following is an interview question from Mark Joshi et al. Quant Job Interview. > Question: Why is Brownian motion useful in finance? I am from a Pure Maths PhD background (functional analysis, particularly Banach Space Theory). I would like to venture into quant finance industry after my PhD graduation. Thus, I have no idea on how to answer question above as it seems that most stochastic calculus books would involve talking about Brownian motion but never give motivations. ## Answer by Kevin (score 19, accepted) https://quant.stackexchange.com/a/50170 Brownian motion is simply the limit of a scaled (discrete-time) random walk and thus a natural candidate to use. It is very intuitive and arguably one of the simplest and best understood time-continuous stochastic processes. Also, don't forget that you obtain many more stochastic processes as functions of a (time-changed) Brownian motion. In many books on stochastic calculus, you first define the Ito integral with respect to a Brownian motion before you extend it to general semimartingales. Assuming that log-returns follow a Brownian motion (with drift), you can easily derive closed-form solutions for option prices. Brownian motion is furthermore Markovian and a martingale which represent key properties in finance. Brownian motion was first introduced by Bachelier in 1900. Samuelson then used the exponential of a Brownian motion (geometric Brownian motion) to avoid negativity for a stock price model. Based on this work, Black and Scholes found their famous formula in 1973. ## Answer by Alex C (score 12) https://quant.stackexchange.com/a/50171 Physical objects move according to simple smooth curves that can be represented by low order polynomials: a straight line, a parabola, an ellipse, etc. Financial market prices move in a completely different way, as can be seen by looking at any graph of stock prices, interest rates etc. in a newspaper: there are constant, erratic fluctuations, sometimes in one direction, sometimes in the other, sometimes small and sometimes big, that give the curve a rough, random appearance. The Brownian Motion is a suitable model for this kind of curve.
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