From Heat to Black Scholes

The temperature change from day-to-day at different soil depths is shown in diagram E. Close to the surface, the heat from the sun causes high variations in temperature from noon to noon. The light blue 5cm sine wave captures the variation. As we go deeper into the soil (20cm & 40cm), the daily sine waves flatten out via a process called exponential decay. Fourier’s key observation in deriving the heat eqn was that it could be represented as sum of these sine waves. In fact, he believed that any function could be represented as a sum of sine or cosine waves. The method became known as Fourier series. It revolutionized the way motion and change were analyzed.

The heat eqn models how temperature T changes as both time t and depth z change. In the late 1960s Black, Scholes and Merton were trying to develop a model that captures how an option value V changes as both time t and the option’s underlying stock price S change. However, unlike the smooth continuous variables z and t in the heat eqn, one of the variables, S, that BS & M were dealing with was random. Solutions to differential eqns were not possible if the variables were random or jagged. They solved this problem by assuming that the random nature of the stock price S could be diversified away by assuming that the option price V was perfectly correlated with the stock price S if the right amount of stock was held. This amount is calculated as dV/dS, aka the stock’s delta. They combined this assumption with an, at the time, new branch of mathematics called Ito calculus to convert the eqn for the random stock price movement, dS, into eqn E below, the BSM eqn for the non-random option price movement, dV.

The similarities btw the BSM eqn and Fourier’s heat eqn are shown with the vertical parallel lines btw eqns E and A. In the heat eqn, the curvature term and its coefficient control us how fast heat flow changes as it goes deeper into the ground. In the BSM eqn, on the other hand, the curvature term controls how fast the option price changes as the stock price and its delta change.

The BSM eqn, like all differential eqns is an eqn of movement. One of its solutions is the formula(G) for the price of a call option at a particular point time for a given stock price, S. To arrive at this formula, BS & M first converted the BSM eqn back into the heat eqn. They then used techniques for solving the heat eqn to derive the call option formula.