Black-Scholes Is the Heat Equation

The heat equation describes temperature $w(z,\tau)$ spreading along a rod:

Heat flows from hot spots to cold ones, and any sharp edge in the starting profile is smoothed out immediately. The Black-Scholes PDE from the previous lesson looks messier:

It has coefficients that grow with $S$, a first-derivative term, a discounting term $rV$, and its condition sits at the end (the payoff at $t = T$) instead of at the start. Each difference goes away with one substitution. Log price $x = \ln S$ makes the coefficients constant, and time to expiry $\tau = T - t$ lets time run forward from the payoff. Multiplying by $e^{r\tau}$ removes the $rV$ term. Moving to a frame that drifts at $r - \tfrac12\sigma^2$ removes the first-derivative term.

What is left is $w_\tau = \tfrac12\sigma^2 w_{zz}$, with the payoff as the starting temperature. So an option price is a payoff that has been left to diffuse for $\tau$ years. A call's payoff has a kink at the strike, and a moment earlier than expiry the price curve is already smooth, just as a rod with one hot half and one cold half has a smooth temperature profile a second after you put them together.

We will use one running example throughout: $S = 100$, $K = 100$, $r = 5\%$, $\sigma = 20\%$, $T = 1$. The Black-Scholes call price is $10.45$, and we will get that number back out of the heat equation.