# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PDS] Introducing Brownian Motion

When we write the stock return process over time-step $\Delta t$ as:

$\Delta S = \mu S \Delta t + \sigma S \epsilon \sqrt{\Delta t}$

then borrowing from classical calculus, we are tempted to take the natural next step and take a continuous time limit $\Delta t \to 0$ and write

$dS = \mu S dt + \sigma S \epsilon \sqrt{dt}$

But we immediately have a problem when we do that. Why?

When $\Delta t \to 0, \sqrt{\Delta t}$ does not. Because when $\Delta t$ is small, $\sqrt{\Delta t}$ is greater than $\Delta t$, i.e. $dt \not \to 0$. (On the other hand, however, as Wilmott (2006), Ch. 4 points out, the term $\sqrt{\Delta t}$ is multiplied by a term $\epsilon \sim \mathcal{N}(0, 1)$ which has a mean zero. So while $\sqrt{\Delta t}$ does not go to $0$, it is multiplied by a time whose average value is $0$.)

So we can’t seem to proceed much with the intuition of classical calculus. And indeed this is the case. When there is randomness involved, rules of classical calculus can no longer be applied, and for that we need a new kind of calculus developed by Kiyoshi Ito, popularly called as Ito or, more generally as, stochastic calculus.

The offending random term, $\epsilon \sqrt{\Delta t}$, it turns out describes the finite time behavior of increments of a continuous time process called the Brownian Motion.

That is, Brownian Motion $X(t)$ is that process whose changes in finite time $\Delta t$ are described by $\epsilon \sqrt{\Delta t}$, that is $\Delta X(t) \sim \epsilon \sqrt{\Delta t}$. This is not a definition of Brownian Motion but it does intuitively capture one of the most important properties of Brownian Motion.

As we discussed today, a simple way of heuristically constructing Brownian Motion is to take a symmetric Random Walk and scale it ‘appropriately’ $(\mbox{as } \sqrt{\frac{t}{n}})$. For more, see  Wilmott (2006), Ch. 4.