In 1789 Joseph Fourier accompanied Napoleon as scientific advisor on an expedition to Egypt. When Fourier came back, he developed the heat equation (A) in the diagram above. An early application of the heat eqn was a model for (B) describing heat flow from a ground surface through soil depth. In the model, dT/dt captures how temperature T changes in the soil as time t passes. dT/dz models the direction of heat flow, where z is depth. And the curvature term tells us how fast that flow changes as it goes deeper into the soil.
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.
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![Here's the alt text for the image: Infographic titled "From Heat to Black Scholes" visually comparing the Heat Equation and the Black-Scholes Equation. Top Section: Heat Equation A: Heat Equation: Displays the partial differential equation dt dT = ρc k dz 2 d 2 T , labeled with terms: "Change in temp" ( dt dT ), "Thermal Diffusivity" ( ρc k ), and "Curvature" ( dz 2 d 2 T ). Arrows point to these components. B: Model for Soil Temp Change: A diagram illustrating heat flow from the ground surface through soil depth (z). It shows temperature T changing with depth z, thermal diffusion, and the direction of heat flow. The curvature of the temperature profile with depth is also indicated. C: Boundary Conditions (Heat Equation): Lists conditions T(0,t)=T AVG +A(0)sin(ωt) and T(∞,t)=T AVG . D: Solution Equation (Heat Equation): Shows the solution T(z,t)=T AVG +A(z)sin[ωt+B(z)]. E: Daily Soil Temperature Change: A line graph showing temperature (in °C) over time (from noon to noon over three cycles) at different soil depths: 5 centimeters (light blue, high amplitude sine wave), 20 centimeters (red, lower amplitude sine wave with a phase shift), and 40 centimeters (green, even lower amplitude sine wave with a larger phase shift). H: Soil Temp Change by Depth: A line graph showing soil depth (in cm) on the y-axis and temperature (in °C) on the x-axis at different times (12:00, 18:00, and 22:00). The curves show how the temperature profile changes with depth throughout the day, flattening out as depth increases. Bottom Section: Black Scholes Equation E (repeated, with a different label): Black Scholes Equation: Displays the partial differential equation dt dV =− 2 1 σ 2 S 2 dS 2 d 2 V −rS dS dV +rV, labeled with terms: "Theta" ( dt dV ), "Vol Scaling" ( 2 1 σ 2 S 2 ), "Gamma" ( dS 2 d 2 V ), "RF Rate * Delta" (rS dS dV ), and "RF Rate * V" (rV). Vertical parallel lines visually connect the "Curvature" term in the Heat Equation to the "Vol Scaling * Gamma" term in the Black-Scholes Equation. F: Boundary Conditions (Black Scholes Equation): Lists conditions V(S,T)=Max(S−K,0) and V(0,t)=0. G: Solution Equation for a Call Option: Shows the Black-Scholes formula for a call option: c=S 0 N(d 1 )−Ke −rT N(d 2 ), where d 1 and d 2 are also provided. I: BSM Option Price Change: A line graph showing BSM option price on the y-axis and stock price on the x-axis for different times to expiration: 1 month (red, steepest curve), 6 months (green, intermediate curve), and 12 months (blue, flattest curve). The option price increases with the stock price and time to expiration. The infographic uses arrows and visual connections to highlight the mathematical similarities between the Heat Equation and the Black-Scholes Equation, particularly the role of the second-order derivative (curvature) in both models. It illustrates the solutions of the Heat Equation with soil temperature changes over time and depth, and the solution of the Black-Scholes Equation with the call option price as a function of the stock price and time to expiration.](https://b2346141.smushcdn.com/2346141/wp-content/uploads/2025/04/From_Heat_to_Black_Sholes.jpeg?lossy=1&strip=1&webp=1)
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.