What is a Wiener Process/Brownian Motion and What are its Uses in Finance?

Short answer

The Wiener process or Brownian motion is a stochastic process with stationary independent normally distributed increments and which also has continuous sample paths. It is the most common stochastic building block for random walks in finance.

Example

Pollen in water, smoke in a room, pollution in a river, are all examples of Brownian motion. And this is the common model for stock prices as well.

Long answer

Brownian motion (BM) is named after the Scottish botanist who first described the random motions of pollen grains suspended in water. The mathematics of this process were formalized by Bachelier, in an option-pricing context, and by Einstein. The mathematics of BM is also that of heat conduction and diffusion.

Mathematically, BM is a continuous, stationary, stochastic process with independent normally distributed increments. If Wt is the BM at time t then for every t, t > 0, Wt+T - Wt is independent of {Wu :0 < u < t}, and has a normal distribution with zero mean and variance t.

The important properties of BM are as follows:

• Finiteness: the scaling of the variance with the time step is crucial to BM remaining finite.

• Continuity: the paths are continuous, there are no discontinuities. However, the path is fractal, and not differentiable anywhere.

• Markov: the conditional distribution of Wt given information up until t< t depends only on WT.

• Martingale: given information up until t< t the conditional expectation of Wt is WT.

• Quadratic variation: if we divide up the time 0 to t in a partition with n + 1 partition points ti = it/n then

• Normality: Over finite time increments ti—1 to ti, Wti — Wti_ 1 is normally distributed with mean zero and variance ti — ti— 1.

You'll see this 'W' in the form dW as the stochastic increment term in stochastic differential equations. It might also appear as dX or dB, different authors using different letters, and sometimes with a time subscript. But these are all the same thing!

It's often easiest just to think of dW as being a random number drawn from a normal distribution with the properties: E[dW] = 0and E[dW2] = dt.

BM is a very simple yet very rich process, extremely useful for representing many random processes especially those in finance. Its simplicity allows calculations and analysis that would not be possible with other processes. For example, in option pricing it results in simple closed-form formulae for the prices of vanilla options. It can be used as a building block for random walks with characteristics beyond those of BM itself. For example, it is used in the modelling of interest rates via mean-reverting random walks. Higher-dimensional versions of BM can be used to represent multi-factor random walks, such as stock prices under stochastic volatility.

One of the unfortunate features of BM is that it gives returns distributions with tails that are unrealistically shallow. In practice, asset returns have tails that are much fatter than those given by the normal distribution of BM. There is even some evidence that the distribution of returns has infinite second moment. Despite this, and the existence of financial theories that do incorporate such fat tails, BM motion is easily the most common model used to represent random walks in finance.

References and Further Reading

Bachelier, L 1995 Theorie de la Speculation. Jacques Gabay

Brown, R 1827 A Brief Account of Microscopical Observations. London

Stachel, J (ed.) 1990 The Collected Papers of Albert Einstein. Princeton University Press

Wiener, N 1923 Differential space. Journal of Mathematics and Physics 58 131-174

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