Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

BR: Brownian Motion – I

BR’s third chapter forms the core of the book. Here they take the intuition developed in ‘Discrete Processes’ to a more formal continuous setting. The chapter can be seen as divided into three parts:

• Brownian Motion
• Ito Calculus
• Change of Measure

In this post we briefly take a look at BR’s treatment of the construction of Brownian Motion (BM).

Looking at a real time series of asset returns, BR argue that BM, prima facie, offers a good way of characterizing it. Not only is BM “…sophisticated enough to produce interesting models, but is simple enough to be tractable” mathematically. Even though a real financial time series is ‘noisier’ than the paths generated by BM, it offers a ‘basis’ to build more sophisticated continuous processes with.

Like in most other introductory textbooks, BR also develop BM from the simple binomial setting. Their treatment is quite concise, and not the most formal (i.e. starting with definitions etc.), but is clear enough to be understandable.

The Binomial Random Walk

A binomial random walk $W_n(t)$ is defined as:

$W_n(\frac{i}{n}) = W_{n}(\frac{i - 1}{n}) + \displaystyle \frac{X_i}{\sqrt{n}}$

where $X_i$ are a sequence of independent binomial random variables taking values $+1$ or $-1$ with equal probability – i.e. after each time step the change in the value of the random walk is either $\frac{1}{\sqrt{n}}$ or $-\frac{1}{\sqrt{n}}$.

Given that the jump sizes are scaled by the inverse of $\sqrt{n}$, BR show that this “…seems to force some kind of convergence” – in that the binomial random walk $W_n(t)$ doesn’t “blow up”. To see this, write:

\begin{aligned} W_n(\frac{nt}{n}) &= W_{n}(\frac{nt - 1}{n}) + \displaystyle \frac{X_{nt}}{\sqrt{n}} \\&= W_{n}(\frac{nt - 2}{n}) + \displaystyle \frac{X_{nt-1}}{\sqrt{n}} + \displaystyle \frac{X_{nt}}{\sqrt{n}}\end{aligned}

Recursively expanding $W_n(\frac{nt - i}{n})$ and using $W_n(0) = 0$ gives:

$W_n(t) = \sqrt{t} \Big(\frac{\displaystyle\sum_{i = 1}^{nt} X_i}{\displaystyle\sqrt{nt}}\Big)$

where we have multiplied both the numerator and denominator by $\sqrt{t}$.

From the central limit theorem, for $n$ large, the term in parentheses above tends to a Gaussian $N(0, 1)$ distribution, implying that the distribution of $W_n(t)$ tends to a Gaussian $N(0, t)$ distribution.

The moral of the story is that at the end of any time $nt$, the distribution of the sum $W_n(t)$ tends to Gaussian – i.e. at the end of any time the distribution of possible values of the random walk is Gaussian. This completes the characterization of the BM in the binomial setting. Let $n \rightarrow \infty$ and we have constructed our BM/Wiener Process in continuous time.

They go and define the continuous BM formally. It’s standard stuff and there are no surprises here. But I like their concise description of the construction of BM – it assumes certain comfort level with probability theory, but is quite succint and to-the-point.

Next: From BM as a ‘basis’ to a stochastic process for asset returns.