Chapter 2: FAQs 95
- Markov: the conditional distribution ofWtgiven
information up untilτ<tdepends only onWτ. - Martingale: given information up untilτ<tthe
conditional expectation ofWtisWτ. - Quadratic variation: if we divide up the time 0 totin
a partition withn+1 partition pointsti=it/nthen
∑n
j= 1(
Wtj−Wtj− 1) 2
→t.- Normality: Over finite time incrementsti− 1 toti,
Wti−Wti− 1 is normally distributed with mean zero
and varianceti−ti− 1.
BM is a very simple yet very rich process, extremely
useful for representing many random processes es-
pecially those in finance. Its simplicity allows calcula-
tions and analysis that would not be possible with other
processes. For example, in option pricing it results in
simple closed-form formulæ 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 stochas-
tic 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 that given by the normal distribution
of BM. There is even some evidence that the distribu-
tion of returns have 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.