The Heat Equation and the Generator of Brownian Motion

The Black-Scholes PDE is:

\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0.

At first glance this looks like an equation from nowhere. But it is, in fact, the heat equation with a coordinate change — the fundamental PDE that describes how temperature diffuses, which is also exactly the equation governing the expected future payoff of an option as a function of current price and time to expiry.

The connection is: if $(W_t)$ is a BM with drift $m$ and variance $\sigma^2$ , and $\varphi(t, x) = \mathbb{E}_x[f(W_t)]$ (the expected value of $f$ at time $t$ , starting from $x$ ), then $\varphi$ satisfies:

\partial_t \varphi = \mathcal{L} \varphi, \quad \text{where} \quad \mathcal{L} = m\partial_x + \frac{\sigma^2}{2}\partial_{xx}.

The operator

\mathcal{L}

is the infinitesimal generator of BM. For driftless BM and

f(x) =

a call payoff, this reduces immediately to the heat equation

\partial_t \varphi = \frac{1}{2}\partial_{xx}\varphi

. The Black-Scholes PDE is this equation in log-price coordinates plus a discounting adjustment.

Learning objectives. By the end of this note you should be able to:

Derive the heat equation $\partial_t p_t = \frac{1}{2}\partial_{xx} p_t$ for the transition density of SBM.
State and apply Theorem 2.10.3: the generator of $d$ -dimensional BM with drift $m$ and covariance $\Gamma$ is $\mathcal{L}f = m \cdot \nabla f + \frac{1}{2}\sum_{jk}\Gamma_{jk}\partial_{jk}f$ .
Verify that $\varphi(t,x) = \mathbb{E}_x[f(W_t)]$ satisfies $\partial_t \varphi = \mathcal{L}_x \varphi$ with initial condition $\varphi(0,x) = f(x)$ .
Connect the generator to the Black-Scholes PDE via a change of coordinates.
Interpret the Chapman-Kolmogorov equation and the Markov semigroup $P_t f(x) = \mathbb{E}_x[f(W_t)]$ .

In this lesson

The Transition Density of Brownian Motion
The Heat Equation: Heuristic Derivation
The Generator of Brownian Motion
The Backward Kolmogorov Equation
Connection to the Black-Scholes PDE
Example: The Call Option Price as a Heat Kernel Convolution

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