Brownian Motion as a Symmetric Random Walk Limit

8/7/2022

Brownian motion is the continuous-time limit of an incredibly simple discrete object: a symmetric random walk that steps $+1$ or $-1$ with equal probability. This post keeps my original notes and adds interactive charts so you can see the convergence happen with your own eyes.

Attention Conservation Notice

The 1st post in a series of 5 in which the Black–Scholes–Merton Model for pricing European options is derived. This follows Stephen Blyth’s An Introduction to Quantitative Finance closely, with embellishment and interactive plots. This is probably:

not very helpful if you pragmatically want to price an option today,
overwhelming if you do not like math,
possibly missing context you might want if you have a lot of mathematical maturity (we will fill in more later).

Controls

Steps (n)

Horizon T

Paths shown

Seed

CLT samples

W_{n,t} samples

Pick t for W_{n,t} (variance should be t)

dt = 0.0010 | t = 0.8000 of T = 1.0000

A Symmetric Random Walk

Let $\{\xi_i\}$ be IID with $\mathbb P(\xi_i = 1) = \mathbb P(\xi_i = -1) = 1/2$ and define the symmetric random walk

S_N \;=\; \sum_{i=1}^{N} \xi_i.

Below is the interactive replacement for the old fake “stonk” chart. It draws multiple paths of $S_N$ .

From this we can eyeball some facts: $\mathbb E[\xi_i]=0$ , $\mathrm{Var}(\xi_i)=1$ , so $\mathbb E[S_N]=0$ and $\mathrm{Var}(S_N)=N$ . The walk meanders, but its variance grows linearly with the number of steps.

The Central Limit Theorem

Lindeberg–Levy CLT

Suppose $\{X_1,\dots,X_n,\dots\}$ are IID with $\mathbb E[X_i]=\mu$ and $\mathrm{Var}(X_i)=\sigma^2 < \infty$ . Then

\sqrt{n}\,(\bar X_n - \mu) \;\xrightarrow{d}\; \mathcal N(0,\sigma^2).

For our $\xi_i$ with $\mu=0$ and $\sigma^2=1$ ,

\frac{1}{\sqrt{n}}\sum_{i=1}^n \xi_i \;\xrightarrow{d}\; \mathcal N(0,1).

The histogram below is an interactive version of the original discussion: it shows the empirical distribution of $\frac{1}{\sqrt{n}}\sum \xi_i$ over many samples, overlaid with the $N(0,1)$ pdf.

By the Lindeberg-Levy CLT, regardless of the step distribution shape (here ±1), the normalized average tends to a standard normal as steps grow.

Applying CLT to our random walks $\xi_i$

We can package the normalized partial sums in a convenient form:

W_n \;=\; \sum_{i=1}^{n} \frac{1}{\sqrt{n}} \,\xi_i.

As $n\to\infty$ , $W_n \xrightarrow{d} \mathcal N(0,1)$ . This was the basic statistical backbone in the original post and it survives intact here.

Time -> Brownian Motion

To go from a one-time statement to a whole process, fix a horizon $T>0$ , let $dt = T/n$ , and define

W_n(t) \;=\; \sum_{i=1}^{\lfloor t/dt \rfloor} \sqrt{dt}\,\xi_i,\quad t\in[0,T].

Each increment has variance approximately equal to the time length of the increment, so for fixed $t$ we get $W_n(t) \xrightarrow{d} \mathcal N(0,t)$ . Functionally,

W_n(\cdot) \;\Rightarrow\; W(\cdot),

where $W$ is Brownian motion (Donsker’s invariance principle).

The plot below compares the empirical distribution of $W_n(t)$ against $N(0,t)$ for your chosen $t$ .

As n grows with step size sqrt(dt), the partial sum at time t has variance t. The small mismatch you may notice for certain t values is the integer rounding in floor(t/dt).

As in the original text, the tiny discrepancy you might see is due to the integer rounding in $\lfloor t/dt \rfloor$ .

Discrete Intuition

Intuitively, Brownian motion is the limit of the symmetric random walk with ever smaller time steps and height scaled by $\sqrt{dt}$ . Equivalently, it is the unique (in distribution) process with independent, stationary normal increments. For any $t,\Delta t>0$ ,

W_{t+\Delta t} - W_t \sim \mathcal N(0,\Delta t).

A 3D View (Augmentation)

The original post ended with 2D plots; here is a purely aesthetic bonus: independent copies on each axis give a Brownian path in $\mathbb R^3$ .

Stay Tuned

In the next post, we will use what we have built up here to start looking at Stochastic Differential Equations and Geometric Brownian Motion. That is where the Black–Scholes–Merton dynamics will emerge.

Extra Notes

Properties of Brownian Motion

a) $W_0 = 0$

b) For $s \ge t$ , $W_s - W_t \sim \mathcal N(0, s - t)$

c) For $s \ge t$ , $W_s - W_t$ is independent of $W_t$ (indeed independent of the past $\mathcal F_t$ )

d) $W$ is a martingale since

\mathbb E\!\left[W_s \mid \mathcal F_t\right] \;=\; \mathbb E\!\left[(W_s - W_t) + W_t \mid \mathcal F_t\right] \;=\; W_t.

Notes

There are many proofs of the Central Limit Theorem. Since we now know Brownian motion is a martingale, it is also worth looking at the Martingale Central Limit Theorem.

Donsker’s invariance principle is the precise theorem that upgrades the one-time CLT to process-level convergence in function space. It is the formal “random walk becomes Brownian motion” statement.

Brownian motion is also called a Wiener Process. Norbert Wiener helped found cybernetics, a field about feedback and communication in complex systems. His book The Human Use of Human Beings is still a spark.

Sources

Mainly Stephen Blyth, An Introduction to Quantitative Finance, chapter 16.
Wikipedia links scattered throughout this post for CLT, martingales, and Donsker’s theorem.