The Four Faces of Brownian Motion

Why does the Black-Scholes formula take the particular form it does — with $e^{-rT}$, $d_1$, $d_2$? The answer lies in four structural properties of Brownian motion that are each exploited at a different step:

• Martingale: under $\mathbb{Q}$, the discounted stock price is a martingale — and BM is the prototype continuous martingale.
• Markov: the future of the log-price depends only on the current log-price — BM restarts fresh after each observation.
• Gaussian: the distribution of $W_T - W_t$ is completely characterised by its mean and variance — allowing exact computation of payoff expectations.
• Self-similar: volatility over $T$ days is $\sigma\sqrt{T}$ — because $W_T \overset{d}{=} \sqrt{T} W_1$.

This note develops all four characterisations rigorously from the definition of BM, and shows how each is used in pricing and risk management.

• Verify that $W_t$ is a continuous martingale and prove that $W_t^2 - t$ and $e^{\theta W_t - \theta^2 t/2}$ are also martingales.
• State the Markov property of BM and the restart property: after time $s$, $\hat{W}_u = W_{s+u} - W_s$ is an independent SBM.
• Define a Gaussian process, identify BM's covariance function $\Gamma_{st} = \min(s,t)$, and use it to compute covariances.
• State Brownian scaling (self-similarity) and derive the square-root-of-time rule for volatility.
• Use these four properties to compute $\mathbb{E}[f(W_T)]$, $\mathbb{E}[W_s W_t]$, and $\mathbb{P}(W_T \ge a)$ for basic functions $f$.

1. Face 1: Brownian Motion as a Continuous Martingale
2. Face 2: Brownian Motion as a Markov Process
3. Face 3: Brownian Motion as a Gaussian Process
4. Face 4: Brownian Motion as a Self-Similar Process
5. Computing Expectations via the Four Faces