# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## BR: Brownian Motion – II

It’s obvious that a simple BM cannot be used as a model for evolution of asset returns. But, as mentiond earlier, it can serve as a useful ‘basis’ to construct more sophisticated stochastic processes. BR first make the reader consider the following scaled BM:

$S_t = \sigma W_t + \mu t$

Let’s call this model the Normal BM. They show/argue that while this scaling (by $\mu$) and shifting (by $\mu t$) allows one to ‘map’ real data better, the simulated sample path have a distrubing property, that the asset values can go negative under this ‘model’. In fact, for any time $T > 0$, there is a positive probability that $S_T < 0$. This is a problem – as asset prices cannot go negative.

The next transformation they consider is:

$S_t = exp(\sigma W_t + \mu t)$

Being an exponential transformation, this model obviates negative asset prices – and turns out is also more realistic in the paths it generates. It happens to be a well known stochastic process and is called the Geometric BM (GBM). Prima facie, GBM does look like a reasonable choice.

It’s good that just a simple exponential transformation of the BM allows us to characterize the asset price process – at least for now. As BR say before moving on to developing the calculus of BMs (stochastic/Ito calculus), “Brownian motion can prove an effective building block”.

Until we have further evidence to the contrary, we will stick to GBM as our model of choice (in general, this is always a good way of building models – start with a simple one that can characterize the most basic features of the data well, and then if it fails (say, we have more information at some point) consider more complex ones. It’s good to remember that models are at best idealizations/abstractions – and (can) never tell the whole story)