The Heat Equation Connection to Black Scholes

The Black-Scholes equation transforms into the heat equation through a change of variables, which is why its formula uses the normal distribution.

The Black-Scholes-Merton partial differential equation transforms directly into the heat equation from physics through a change of variables, and this connection is why the original 1973 derivation could borrow a known closed-form solution instead of solving a new equation from scratch. Interviewers ask about the transformation because it tests whether a candidate understands the mechanics of the model rather than having only memorized the final formula.

The equation for an option price V as a function of time t and stock price S states that the partial derivative of V with respect to time, plus one half sigma squared times S squared times the second partial of V with respect to S, plus r times S times the first partial of V with respect to S, minus r times V, equals zero. This equation has variable coefficients, since the terms multiplying the derivatives depend on S itself, which makes it look nothing like the constant-coefficient heat equation at first glance.

The terminal condition is the option payoff at maturity, for example the maximum of S minus the strike and zero for a call, and the equation is solved backward from maturity to today, which is itself a clue that a change of variables running time backward and rescaling the price axis logarithmically will simplify things.

Substituting x equal to the logarithm of S removes the variable coefficients that depend on S, because the logarithmic transformation converts multiplicative moves in the stock price into additive moves in x, exactly the structure geometric Brownian motion has under Ito's lemma. A further substitution reversing the direction of time, replacing t with the time remaining until maturity, removes the backward character of the original equation and turns it into a forward-in-time problem. After both substitutions and an additional exponential rescaling to absorb the discounting term, the equation reduces exactly to the standard heat equation: the partial derivative with respect to the new time…

This is the same equation that describes how heat diffuses through a rod over time, with the option's transformed price playing the role of temperature and the transformed stock price playing the role of position along the rod. The physical analogy is not decorative. The heat equation had a well-known solution using the Gaussian kernel decades before 1973, and recognizing the Black-Scholes equation as a disguised heat equation meant the solution could be written down directly rather than derived from scratch.