Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Brownian Motion and Random Walk Models › The Simple Random Walk

Michael BrenndoerferNovember 22, 202510 min read

Quantitative FinanceAll chapters

Part 5 of 15Stochastic Calculus and Pricing Theory

Chapter 26Brownian Motion and Random Walk Models › The Simple Random Walk

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A random walk is exactly what it sounds like: a sequence of steps where each step's direction is determined by chance. The simplest version uses fair coin flips to decide whether to step up or down. Before diving into the formal mathematics, consider the intuition behind this construction. Imagine you are standing at the origin of a number line. You flip a fair coin repeatedly: if it lands heads, you take one step to the right; if tails, one step to the left. After many flips, where do you end up? This deceptively simple question reveals important properties of randomness and diffusive processes.

The random walk is our conceptual foundation because it distills randomness to its purest form. Each step is completely independent of every previous step, yet when we observe the accumulated position over time, patterns emerge. The walker tends to drift away from the origin, not because of any systematic force, but simply because random movements accumulate. This phenomenon, where independent random increments sum to produce gradually increasing uncertainty, lies at the heart of all diffusion processes in physics, biology, and finance.

Simple Random Walk

A simple random walk is a discrete-time stochastic process $\{S_n\}_{n=0}^{\infty}$ where $S_0 = 0$ and:

S_n = S_{n-1} + X_n = \sum_{i=1}^{n} X_i

where:

$S_n$ : position of the random walk after $n$ steps
$S_{n-1}$ : position after $n-1$ steps
$X_i$ : independent step at time $i$ , taking values $+1$ or $-1$ with probability $1/2$
$n$ : current time step

The definition above encapsulates several important ideas. First, the recursive formulation $S_n = S_{n-1} + X_n$ tells us that the position at any time depends only on the previous position plus a new random increment. This is the Markov property in its simplest form: the future depends on the present but not on the past. Second, the alternative representation as a sum $S_n = \sum_{i=1}^{n} X_i$ emphasizes that the current position is simply the cumulative total of all steps taken so far. This summation viewpoint connects directly to the Central Limit Theorem and explains why random walks eventually produce Gaussian distributions.

Starting at zero, after $n$ steps the walker's position is the sum of $n$ independent coin flips, each contributing $+1$ (heads) or $-1$ (tails). Let's simulate this process:

In[2]:

Code

import numpy as np


def simple_random_walk(n_steps, n_paths=1):
    """Simulate simple random walks with +1/-1 steps."""
    steps = np.random.choice([-1, 1], size=(n_paths, n_steps))
    positions = np.cumsum(steps, axis=1)
    positions = np.hstack([np.zeros((n_paths, 1)), positions])
    return positions

Out[3]:

Visualization

Line chart showing five random walk paths diverging from zero over 1000 time steps. — Simulated paths of five simple random walks over 1000 discrete time steps. All paths begin at the origin and evolve through independent unit steps (plus or minus 1), showing how random fluctuations accumulate to produce divergent trajectories.

The paths exhibit several notable features. They wander away from zero but occasionally return. Some paths drift predominantly positive or negative over this sample, though with infinite time, a random walk returns to zero infinitely often (a property called recurrence). The paths look "rough" with jagged local movements but smoother trends at larger scales.

Properties of the Simple Random WalkLink Copied

The mathematical properties of the simple random walk follow directly from the properties of sums of independent random variables, which we covered in the chapter on probability fundamentals. Understanding these properties builds the intuition we need for continuous-time models. Each property tells us something essential about how randomness accumulates over time.

Expected position: The first question we ask about any random process is: what is its average behavior? For a random walk, this amounts to asking where we expect the walker to be after $n$ steps. Since each individual step has an expected value of zero (heads contributes $+1$ and tails contributes $-1$ , each with probability $1/2$ , so $\mathbb{E}[X_i] = (1/2)(+1) + (1/2)(-1) = 0$ ), and since expectation is linear, the expected position after $n$ steps must also be zero:

\begin{aligned} \mathbb{E}[S_n] &= \mathbb{E}\left[\sum_{i=1}^{n} X_i\right] && \text{(definition of } S_n \text{)} \\ &= \sum_{i=1}^{n} \mathbb{E}[X_i] && \text{(linearity of expectation)} \\ &= 0 && \text{(since } \mathbb{E}[X_i] = 0 \text{)} \end{aligned}

where:

$\mathbb{E}[\cdot]$ : expected value operator
$S_n$ : position after $n$ steps
$X_i$ : random step at index $i$

This result, while mathematically straightforward, has a counterintuitive interpretation. The random walk has no systematic drift: on average, the walker stays at the origin. The coin is fair, so there is no bias pushing the walker in either direction. Yet this does not mean the walker stays near zero. As we shall see, the walker typically wanders quite far from the origin, even though the average position remains zero.

Variance of position: To understand how far the walker typically wanders, we need to compute the variance. Each step has variance $\text{Var}(X_i) = \mathbb{E}[X_i^2] - (\mathbb{E}[X_i])^2 = 1 - 0 = 1$ . The variance equals one because the step is either $+1$ or $-1$ , so the squared step is always $1$ , and the squared mean is zero. Since the steps are independent, a fundamental property of variance tells us that the variance of a sum equals the sum of the variances:

\begin{aligned} \text{Var}(S_n) &= \sum_{i=1}^{n} \text{Var}(X_i) && \text{(independence of steps)} \\ &= \sum_{i=1}^{n} 1 && \text{(variance of single step)} \\ &= n && \text{(sum of } n \text{ ones)} \end{aligned}

where:

$\text{Var}(\cdot)$ : variance operator
$S_n$ : position after $n$ steps
$X_i$ : random step variable (variance is 1)
$n$ : total number of steps

This reveals a fundamental insight: while the expected position stays at zero, the typical deviation from zero grows like $\sqrt{n}$ . The standard deviation of the position is $\sqrt{\text{Var}(S_n)} = \sqrt{n}$ , meaning that after 100 steps, the walker is typically about 10 units away from the origin. After 10,000 steps, the typical distance grows to about 100 units. The walker doesn't systematically go anywhere, but uncertainty about its location increases over time. This square-root growth of uncertainty is a universal feature of diffusive processes, appearing in contexts ranging from heat conduction to portfolio theory.

Out[4]:

Visualization

Fan plot showing random walk paths spreading out within square-root-of-time envelope bounds. — Growth of uncertainty in simple random walks over 500 steps. The standard deviation of the position increases with the square root of time (dashed boundary), enveloping the central region around zero while simulated paths spread outward and illustrate the diffusive nature of the process.

Distribution of position: Beyond the mean and variance, we can characterize the entire probability distribution of the walker's position. By the Central Limit Theorem, for large $n$ , the sum $S_n$ of $n$ independent, identically distributed random variables approaches a normal distribution:

S_n \xrightarrow{d} N(0, n)

where:

$S_n$ : position after $n$ steps
$\xrightarrow{d}$ : convergence in distribution
$N(0, n)$ : normal distribution with mean 0 and variance $n$

The Central Limit Theorem is a fundamental result in probability theory. It tells us that regardless of the distribution of individual steps (as long as they have finite mean and variance), their sum becomes approximately Gaussian. For our random walk with $\pm 1$ steps, the convergence is rapid: even after just 30 or 40 steps, the distribution is nearly indistinguishable from a normal distribution. This explains why Gaussian distributions appear so frequently in nature and finance: whenever a quantity is the sum of many small independent contributions, normality emerges.

Let's verify these properties empirically:

In[5]:

Code

import numpy as np

# Simulate many random walks to check distribution
n_paths = 10000
n_steps = 100

walks_many = simple_random_walk(n_steps, n_paths)
final_positions = walks_many[:, -1]  # Position at step 100

# Calculate empirical statistics
theoretical_mean = 0.0
empirical_mean = final_positions.mean()
empirical_std = final_positions.std()
theoretical_std = np.sqrt(n_steps)

Out[6]:

Console

After 100 steps across 10,000 simulations:
  Theoretical mean: 0.000
  Empirical mean:   -0.004
  Theoretical std:  10.000
  Empirical std:    9.980

The empirical mean is close to the theoretical value of zero, and the standard deviation matches the square root of the number of steps (10). This confirms the random walk's diffusive nature.

Out[7]:

Visualization

Histogram of random walk endpoints with overlaid normal distribution curve. — Empirical distribution of random walk positions after 100 steps compared with the theoretical normal density. Integer-centered bins respect the walk's two-unit lattice spacing, and the resulting histogram closely follows the Gaussian curve, showing the Central Limit Theorem in action.

The empirical distribution matches the theoretical normal distribution well, confirming that the sum of many independent random steps produces Gaussian outcomes.

Independent IncrementsLink Copied

Random-walk increments over non-overlapping time intervals are independent. This property becomes one of the defining characteristics of Brownian motion and directly affects financial modeling. To understand what this means, consider two time intervals: from time $m$ to time $n$ , and from some earlier time $0$ to time $m$ (where $m < n$ ). The change in position from time $m$ to time $n$ is:

S_n - S_m = \sum_{i=m+1}^{n} X_i

where:

$S_n$ : position at time $n$
$S_m$ : position at time $m$ (where $m < n$ )
$X_i$ : independent step at time $i$

This increment depends only on steps $X_{m+1}, \ldots, X_n$ , which are independent of all earlier steps $X_1, \ldots, X_m$ . The steps in the future interval have nothing to do with the steps in the past interval because each coin flip is completely independent of all others. Therefore, knowing where the walker was at time $m$ tells you nothing about how far it will move between $m$ and $n$ .

Independent increments impose a direct restriction on financial models. If price changes follow a random walk, then past price movements cannot predict future movements: the process is a martingale with respect to its natural filtration. In practical terms, this means that technical analysis based on historical price patterns should not provide systematic profits if prices truly follow a random walk. The market incorporates all past information into the current price, and future price changes depend only on new information, which by definition cannot be predicted from the past.