The Heat Equation and Diffusion

Drop a hot poker into one end of a long metal bar and leave it. The heat doesn't jump around randomly — it spreads out smoothly, always from hot toward cold, faster where the temperature changes sharply and slower where it's already nearly flat. There's a single equation describing this, the heat equation, and it turns out to be one of the most reused pieces of mathematics in finance: the Black-Scholes equation is, after renaming the variables, exactly the heat equation. Understanding heat spreading through metal is understanding how an option's value spreads and smooths across possible stock prices.

The analogy: a rumour smoothing through a crowd

Imagine a piece of gossip's "intensity" varying across a crowd standing in a line — very intense right where it started, fading out toward the edges of the crowd who haven't quite heard it clearly yet. Over time, people compare notes with their immediate neighbours, and the sharp spike smooths into a gentle bump, then a nearly flat murmur across everyone. Crucially: the rate at which any one person's sense of the rumour changes depends on how different their neighbours' impressions are from their own — if you're between two people with a much stronger sense of it than you, you catch up fast; if everyone around you already agrees, nothing changes. That "catch up based on how different you are from your immediate neighbours" rule is precisely what the heat equation encodes, with temperature standing in for rumour intensity.

Writing it down

Let $u(x,t)$ be the temperature at position $x$ along the bar, at time $t$. The heat equation says:

$$\frac{\partial u}{\partial t} = \alpha \, \frac{\partial^2 u}{\partial x^2} .$$

In words: the rate temperature is changing over time at a point ($\partial u/\partial t$) equals a constant $\alpha$ (how fast heat conducts through this particular material) times the curvature of the temperature profile at that point ($\partial^2 u/\partial x^2$, the second derivative in space). Curvature is exactly "how different you are from your immediate neighbours' average" — a sharp spike has huge curvature (you're much hotter than both neighbours) and cools fast; a flat, already-even region has zero curvature and stops changing.

This single equation forces heat to always flow from hot to cold, never the reverse, and it forces sharp features to smooth out monotonically — heat never spontaneously re-sharpens into a spike. Both properties matter for finance: an option's payoff typically starts with a sharp kink (a call option's payoff has a corner exactly at the strike), and pricing it backward in time under the Black-Scholes PDE is mathematically the same operation as running the heat equation forward in time on that kink — which is precisely why an option's price is a smooth curve even though its payoff at expiry is not.

The connection is made exact by a change of variables: writing the Black-Scholes PDE in terms of $x=\log(S/K)$ (log-moneyness) and reversing the direction of time turns

$$\frac{\partial V}{\partial t} + \tfrac12\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$

into the plain heat equation with $\alpha = \tfrac12\sigma^2$. In words: volatility $\sigma$ plays the role of the heat-conduction constant — a highly volatile stock is a highly conductive material, spreading and smoothing the payoff's kink fast; a low-volatility stock conducts slowly, leaving the kink comparatively sharp even close to expiry.

Each path here is one random walker; the distribution across many walkers at a fixed time is exactly the heat profile $u(x,t)$. Redraw with a higher volatility (a faster-diffusing "material") and watch the spread of endpoints widen faster — that widening spread is the heat equation's smoothing, viewed one random path at a time instead of as a smooth density.

Worked example 1: a discrete bar, three points, one step

Take a bar with just three points, temperatures $u = (100, 0, 0)$ at time 0 (hot at the left end, everything else cold), spacing $\Delta x = 1$, and a discrete stand-in for the second derivative: $u''_i \approx u_{i-1} - 2u_i + u_{i+1}$.

At the middle point (index 2): $u''_2 = u_1 - 2u_2 + u_3 = 100 - 0 + 0 = 100$. With $\alpha=0.1$ and a time step $\Delta t = 0.05$, the update is $u_2^{\text{new}} = u_2 + \alpha\,\Delta t\, u''_2 = 0 + 0.1(0.05)(100) = 0.5$.

At the left point, treating it as fixed (a boundary held at 100, the usual convention) it stays $100$. At the right point (index 3, boundary held at $0$) it stays $0$.

So after one step, $u = (100, 0.5, 0)$ — heat has begun leaking from the hot end into its immediate neighbour, at a rate set exactly by the curvature computed above, and it has not yet touched the far point, matching the everyday intuition that heat spreads outward one step of contact at a time.

Worked example 2: an option's kink smoothing near expiry

A call option with strike $K=100$ has payoff $\max(S-K,0)$ at expiry — a sharp corner exactly at $S=100$: value is $0$ just below and rises linearly just above. One day before expiry, with volatility $\sigma=30\%$ annualised, the "diffusion distance" heat travels is proportional to $\sigma\sqrt{\Delta t}$ where $\Delta t = 1/252$: $0.30 \times \sqrt{1/252} = 0.30 \times 0.063 = 0.0189$, about 1.9% in log-price terms.

That means the sharp corner at $S=100$ gets smoothed over a band roughly $\pm1.9\%$ wide — from about $98.1 to $101.9 — where the option's value one day out is a rounded curve rather than the payoff's hard kink. Compare a longer-dated option, say 30 days to expiry: diffusion distance scales with $\sqrt{\Delta t}$, so it grows to $0.30\times\sqrt{30/252}=0.30\times0.345=0.103$, over 10%. The kink is smoothed over a far wider band the more time (equivalently, the more "heat conduction time") remains — exactly why deep options trade with rounded, gently curved prices near the strike while very-short-dated options at the money show almost a visible corner.

What this means in practice

• It explains why option prices are always smoother than their payoffs. Any amount of remaining time acts like heat-conduction time, rounding off every kink in the terminal payoff — barrier and digital payoffs, with far sharper kinks, show this rounding even more dramatically near their trigger levels.
• It's why numerical PDE solvers for options borrow directly from heat-transfer numerics. Finite-difference option pricers use the same stability conditions (the ratio of time step to squared space step must stay bounded) that engineers use for heat-transfer simulations, because it is the identical equation.
• It connects to probability through Feynman-Kac. The heat equation's solution can be written as an average over random paths — an early version of the Feynman-Kac link between PDEs and expectations that Black-Scholes pricing depends on directly.