PDEs in Finance: Solving the Black-Scholes Equation
If you have ever cooked something on a metal pan, you already understand partial differential equations. No, seriously. The way heat flows from the burner through the pan to your food follows the exact same type of math that prices options on Wall Street. Chapter 6 of Wilmott’s book makes this connection explicit, and honestly it makes the whole thing feel a lot less scary.
The Black-Scholes Equation Is Just a Heat Equation in Disguise
The Black-Scholes equation is what mathematicians call a parabolic partial differential equation. It has a second derivative with respect to the stock price S and a first derivative with respect to time t. Equations like this have been around since the early 1800s, long before anyone thought about pricing options.
The simplest version is the heat equation:
du/dt = d2u/dx2
Where u is temperature, x is position along a bar, and t is time. The idea is simple: heat flows from hot spots to cold spots, and the rate of flow depends on the temperature gradient. The second derivative captures how “curved” the temperature profile is. If there is a spike, heat flows away from it. If there is a dip, heat flows toward it. The result is that everything smooths out over time.
The Black-Scholes equation works the same way. Any kink or discontinuity in the option payoff at expiry gets smoothed out as you move backward in time. That hockey-stick shape of a call option payoff? It becomes a smooth curve the moment you look at it before expiry.
The Three Pieces of Black-Scholes
Wilmott breaks the Black-Scholes equation into three physical components, and this is genuinely helpful for building intuition.
Diffusion term (the second derivative in S): This is the smoothing part. It acts like heat diffusion, spreading out any sharp features in the option price. In the Black-Scholes equation, the diffusion coefficient depends on S itself, so it is like diffusion through a non-uniform medium.
Convection term (the first derivative in S): Think of this as a breeze. If smoke is diffusing in the air, the wind pushes it in a preferred direction. In the option pricing context, the risk-free rate creates this drift effect, pushing the option value in a particular direction.
Reaction term (the term proportional to V): This is like radioactive decay. It causes the option value to shrink over time, reflecting the time value of money. If you are going to receive a payoff in the future, its present value is less than the payoff itself.
Put them all together and you get a reaction-convection-diffusion equation. Wilmott points out that an almost identical equation describes pollutant spreading along a flowing river with absorption by the sand. The spreading is diffusion, the flow is convection, and the absorption is reaction. Same math, completely different problem.
Boundary and Final Conditions
An equation by itself is not enough. You need to tell it how to start and where the edges are.
For the Black-Scholes equation, we specify a final condition (the payoff at expiry) and boundary conditions (what happens when S = 0 and when S goes to infinity). The equation is “backward” in time, meaning we know what happens at the end (the payoff) and solve backward to find today’s price.
One nice property: the equation is linear. This means if you have two solutions, you can add them together to get a third valid solution. This superposition principle is extremely useful. It breaks down for some more complicated problems (like American options, which we will get to later), but for basic European options it makes life much easier.
Solution Methods
Wilmott covers several approaches, but is refreshingly honest that most of them are not what practitioners actually use day to day.
Transformation to a simpler equation
You can change variables to turn the Black-Scholes equation into the basic constant-coefficient diffusion equation:
dU/dt = (1/2) d2U/dx2
This is done by substituting x = log(S) and doing some algebraic cleanup. The resulting equation is simpler and sometimes useful for finding exact solutions or running simple numerical schemes.
Green’s functions
The Green’s function is a special solution that starts as a spike (a Dirac delta function) at one point and then diffuses outward. It is the fundamental building block. Because of the superposition principle, you can write the solution for any payoff as an integral of the Green’s function weighted by the payoff:
V(S, t) = integral of [Green’s function x Payoff(S’)] dS’
This is exactly how the Black-Scholes formula for calls and puts is derived (more on that in the next chapter). The Green’s function acts like a “response kernel” that tells you how each piece of the payoff contributes to today’s price.
Fourier series
When you have boundary conditions at two specific asset values (say, the option becomes worthless if the stock goes above or below certain levels), you can expand the solution as a sum of sine and cosine functions. Each term decays at a different rate. This is the Fourier series method, and while it is mathematically elegant, it can have issues with discontinuous payoffs (something called Gibbs phenomenon, where the series overshoots near jumps).
Similarity reductions
Sometimes the solution depends on the two variables x and t only through a specific combination of them, like x/sqrt(t). When this happens, the two-variable PDE reduces to a one-variable ordinary differential equation, which is much easier to solve. This trick shows up repeatedly in later chapters for more complicated problems.
The Real Answer: Numerical Solutions
Here is the practical truth that Wilmott emphasizes: in the vast majority of real-world cases, you solve the Black-Scholes equation numerically. The analytical solutions are nice for simple cases, but the moment you add realistic features like discrete dividends, barriers, or early exercise, exact formulas disappear.
The good news? Parabolic PDEs are among the easiest equations to solve numerically. You discretize the asset price and time into a grid, and then step backward from the known payoff at expiry. Even the simplest finite difference methods work well. Wilmott promises to cover these in detail later (Chapters 15, 77, and 78) and essentially asks the reader to trust that every PDE derived in the book will eventually be solved numerically.
Why This Matters
Understanding that the Black-Scholes equation is “just” a diffusion equation has real practical value:
Smoothing: Any sharp feature in the payoff gets diffused away before expiry. This is why option prices are smooth curves even when payoffs have kinks.
Uniqueness: As long as the solution does not grow too fast for large S, there is exactly one answer. You do not have to worry about multiple solutions.
Numerical tractability: Diffusion equations are well-understood and easy to solve on a computer. Decades of engineering experience with heat equations directly applies.
Physical intuition: Thinking about option pricing as heat flow or pollutant spreading can help you guess the qualitative behavior before doing any calculations.
The chapter is intentionally short and serves as a primer. Wilmott keeps the entry requirements low because the deep understanding of PDEs is not required to follow the rest of the book. What matters is recognizing the type of equation, knowing it has nice properties, and trusting that numerical methods will handle the heavy lifting.
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