94 Frequently Asked Questions In Quantitative Finance
What is Brownian Motion and What
are its Uses in Finance?
Short Answer
Brownian Motion is a stochastic process with station-
ary 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 math-
ematics of BM is also that of heat conduction and
diffusion.
Mathematically, BM is a continuous, stationary, stochas-
tic process with independent normally distributed incre-
ments. IfWtis the BM at timetthen for everyt,τ≥0,
Wt+τ−Wtis independent of{Wu:0≤u≤t}, and has a
normal distribution with zero mean and varianceτ.
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.
