Brownian Motion and Its use in Finance
Brownian Motion is a stochastic process with stationary independent normally distributed increments and which has continuous sample paths. It is most common stochastic building block for random walks in finance. Eg of Brownian Motion are Pollen in water, smoke in room , Stock prices are also consider to have Brownian Motion.
Brownian Motion were described from random motions of pollen grains suspended in water. It was first used in option market by Bachelier.
Mathematically, BM is a continuous, stationary , stochastic process with independent normally distributed increments. If Wt is BM at time t then for every t, T>= 0 , (Wt+T – Wt ) is independent of {Wu:0 =< u =< t}, and has normal distribution with zero mean and variance T.
Properties of BM
Continuity: Paths are continuous for BM, there are no discontinuities. However, path are fractal and not differentiable anywhere.
Markov: the conditional distribution of Wt, given information up until T < t depends only on Wt.
BM is very simple yet very rich process, extremely useful for representing random processes especially those in finance. Its simplicity allows calculation and analysis that would not be possible with other process. 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 unfortunate features of BM is that it gives returns distributions with tails that are unrealistically shallow. In practice, asset return have tails that are much fatter than that given by normal distribution of BM. There is even some evidence that the distribution of return have infinite second moment. Despite this, and the existence of financial theories that do not incorporate such fat tails, BM motion is easily the most common model used to represent random walk in finance.