Random Walks

A sequence of iid $\pm 1$ steps whose position after $n$ steps has mean $0$ and variance $n$.

1. The idea

A symmetric simple random walk sets $S_{0} = 0$ and $S_{n} = \sum_{i = 1}^{n} X_{i}$ , where each $X_{i}$ is independently $+ 1$ or $- 1$ with probability $1/2$ each. It is the discrete-time ancestor of Brownian motion.

Because the steps are iid with mean $0$ and variance $1$ , linearity of expectation and independence give $E [S_{n}] = 0$ and $Var (S_{n}) = n$ immediately — no need to enumerate paths.

Random walks are also martingales: $E [S_{n + 1} ∣ S_{n}] = S_{n}$ . This property is what makes gambler's ruin and hitting-probability arguments work by first-step analysis.

2. Why it matters

Random walks are the simplest model of cumulative random P&L or price moves, and the martingale property is the seed idea behind no-arbitrage pricing and stopping-time arguments used throughout the derivatives section.

3. Core mathematics

Definition

S_{n} = i = 1 \sum n X_{i}, X_{i} \sim ii d {- 1, + 1} each w.p. 1/2

Mean

E [S_{n}] = 0

Variance

Var (S_{n}) = n

Martingale property

E [S_{n + 1} ∣ S_{1}, \dots, S_{n}] = S_{n}

Scaling

\frac{S _{n}}{n} d N (0, 1)

CLT gives the Brownian motion limit

4. How to recognize it

Think of this when…

Steps that are iid, mean-zero, symmetric $\pm 1$ (or similar) moves
Questions about the expected position, variance, or typical magnitude after $n$ steps
Setup for a first-step / recursive argument about hitting a boundary
Precursor language to gambler's ruin or Brownian motion questions

5. Worked example

Worked example

A symmetric random walk takes $100$ steps. Find $E [S_{100}]$ , $Var (S_{100})$ , and estimate $P (S_{100} > 10)$ .

1
Each step has mean $0$ , variance $1$ , and steps are independent, so use linearity directly.
$E [S_{100}] = 0, Var (S_{100}) = 100 \times 1 = 100$
2
By CLT, $S_{100} \approx N (0, 100)$ , so $σ = 10$ .
3
Standardize the threshold and use the normal tail.
$P (S_{100} > 10) \approx P (Z > 1) \approx 0.1587$
Answer: $E [S_{100}] = 0$ , $Var (S_{100}) = 100$ , and $P (S_{100} > 10) \approx 0.159$ .

6. Common mistakes

Computing variance of a sum of dependent-looking terms without checking the steps are actually independent
Forgetting that $Var (S_{n}) = n$ only holds when each step has variance $1$ ; rescale if step size differs
Confusing the random walk's martingale property with the (false) idea that $S_{n}$ can't drift far from $0$
Applying the CLT approximation for small $n$ where it is inaccurate

7. Quick check

For a symmetric $\pm 1$ random walk, what is $Var (S_{64})$ ?

Which property directly justifies $E [S_{n + 1} ∣ S_{n}] = S_{n}$ ?

8. Practice

11 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Recursive Expectation / First-Step Analysis Symmetry

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