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Why Black–Scholes and Mass-Balance Equations Share a PDE Form

Article Quant Q&A · Author: user98
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Summary

The document addresses why the Black–Scholes pricing equation resembles a mass or material balance equation, including the advection–diffusion and heat equations. It explains that a change of variables can transform Black–Scholes into a heat equation, which accounts for the shared mathematical structure and provides a route to an analytical solution. The accepted response cautions that mathematical similarity does not establish a direct physical equivalence: material balance equations are typically forward equations, while Black–Scholes is treated as a backward equation.

A second response offers an informal interpretation of the second derivative with respect to the underlying price. In a heat equation, curvature corresponds to local irregularities that diffuse over time; in option pricing, curvature is related to how option value responds nonlinearly to the underlying. That analogy is suggestive rather than a rigorous derivation of investor risk aversion or pricing behavior. The discussion is brief and does not develop a detailed physical mapping between the two systems.

Key ideas

  • A variable transformation can convert the Black–Scholes PDE into a heat equation.
  • The shared PDE form supports mathematical methods, but does not by itself imply the same physical mechanism.
  • Mass-balance equations are generally forward equations, whereas Black–Scholes is a backward pricing equation.
  • The second derivative measures curvature in option value, with a loose analogy to diffusion of local irregularities.

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OptionsDerivatives pricingStatistics
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# Answer by mepuzza (score 1, accepted)


# Can we explain physical similarities between Black Scholes PDE and the Mass Balance PDE (e.g. Advection-Diffusion equation)?












Both the Black-Scholes PDE and the Mass/Material Balance PDE have similar mathematical form of the PDE which is evident from the fact that on change of variables from Black-Scholes PDE we derive the heat equation (a specific form of Mass Balance PDE) in order to find analytical solution to the Black-Scholes PDE.

I feel there should be some physical similarity between the two phenomenon which control these two analogous PDE's (i.e. Black-Scholes and Mass/Material Balance). My question is whether you can relate these two phenomena physically through their respective PDE's? I hope my question is clear, if not please let me know. Thanks.

## Answer by mepuzza (score 1, accepted)

https://quant.stackexchange.com/a/3109

Physical equations tend to be forward equations, whereas in finance one deals with backward equations (e.g. Black-Scholes), so in my opinions analogies are a bit hard to make. The similarity is in the maths that you use, i.e. the PDE you need to solve.

## Answer by Andrew Dabrowski (score 0)

https://quant.stackexchange.com/a/9268

I'm not sure this is what you were getting at, but I think the connection to the heat equation might be explainable. The second derivative $\partial^2 V\over\partial S^2$ provides a measure of the divergence of $V$ from linearity in $S$. In the heat equation that represents a local anomaly which will be smoothed out in time; in the BS pde one could see it as measuring the effect of risk aversion on pricing. I.e., if investors have no risk aversion we would expect $V$ to be (locally) linear in $S$, but risk aversion causes them to weigh rises in $S$ less than falls. Reinterpreting this in terms of risk-neutral probabilities, the option pricing is affected by these modified expectations.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.