Brownian Martingales

Key martingales built from Brownian motion that underlie option-pricing and stopping-time arguments.

1. The idea

Brownian motion itself, $B_{t}$ , is a martingale: $E [B_{t} ∣ F_{s}] = B_{s}$ for $s < t$ , since future increments are independent and mean zero.

$B_{t}^{2} - t$ is also a martingale. This compensates for the fact that $B_{t}^{2}$ alone grows in expectation (since $E [B_{t}^{2}] = t$ ); subtracting $t$ removes that drift and is the key identity behind quadratic-variation arguments.

The exponential martingale $M_{t} = exp (σ W_{t} - \frac{1}{2} σ^{2} t)$ is a martingale for any constant $σ$ , and is the building block of the Girsanov change-of-measure technique used to price derivatives under a risk-neutral measure.

2. Why it matters

These three martingales are the standard toolkit for solving expected-value and stopping-time problems involving Brownian motion, including deriving the Black-Scholes formula and computing hitting-time probabilities via optional stopping.

3. Core mathematics

Brownian motion is a martingale

E [B_{t} ∣ F_{s}] = B_{s}, s < t

Quadratic martingale

E [B_{t}^{2} - t ∣ F_{s}] = B_{s}^{2} - s

Exponential martingale

M_{t} = exp (σ W_{t} - \frac{1}{2} σ^{2} t), E [M_{t} ∣ F_{s}] = M_{s}

Optional stopping (informal)

E [M_{τ}] = M_{0}

for a bounded or suitably regular stopping time $τ$

4. How to recognize it

Think of this when…

Questions asking to verify or use the fact that a given process is a martingale
Hitting-time or stopping-time problems on Brownian motion, solved via optional stopping
Expressions like $exp (σ W_{t} - σ^{2} t /2)$ appearing in a pricing or expectation problem
"Find $E [B_{τ}]$ " or "find $E [B_{τ}^{2}]$ " for a stopping time $τ$

5. Worked example

Worked example

Let $τ$ be the first time Brownian motion hits level $a > 0$ or $- b < 0$ (a two-sided stopping time). Use the martingale property of $B_{t}$ to find $P (B_{τ} = a)$ .

1
Since $B_{t}$ is a martingale and $τ$ is a bounded (a.s. finite, integrable) stopping time, optional stopping gives $E [B_{τ}] = B_{0} = 0$ .
2
At the stopping time, $B_{τ}$ takes only two values: $a$ with probability $q$ , or $- b$ with probability $1 - q$ .
3
Set the expectation equal to zero and solve for $q$ .
$q a - (1 - q) b = 0 \Rightarrow q (a + b) = b \Rightarrow q = \frac{b}{a + b}$
Answer: $P (B_{τ} = a) = \frac{b}{a + b}$ .

6. Common mistakes

Applying optional stopping to an unbounded stopping time without checking integrability conditions
Forgetting to subtract $t$ when using $B_{t}^{2}$ as a martingale (using $B_{t}^{2}$ alone is not one)
Misremembering the exponential martingale's exponent sign or the factor of $1/2$ in front of $σ^{2} t$
Confusing a martingale (constant expectation) with a process that is literally constant

7. Quick check

Which of these is a martingale for standard Brownian motion $B_{t}$ ?

For the exponential martingale $M_{t} = exp (σ W_{t} - σ^{2} t /2)$ , what is $E [M_{t}]$ ?

8. Practice

6 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Recursive Expectation / First-Step Analysis Conditioning on the First Event

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