# 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 a 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 phenomena which control these two analogous PDEs (i.e. Black-Scholes and Mass/Material Balance). My question is, can one relate these two phenomena physically through their respective PDEs? I hope my question is clear, if not please let me know. Thanks.

*PDE=Partial Differential Equation

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I don't understand what you mean by "physical". Black-Scholes is about option pricing whereas a diffusion equation describes a physical process. If mathematically you can reduce one PDE to the other it doesn't mean that the two phenomena are necessary linked. – DaniH Mar 22 '12 at 6:40
DaniH: That's what I am trying to ask if there is really indeed a physical/conceptual connection behind the two PDE's for two different fields. – S_H Mar 22 '12 at 16:48
I think DaniH's point isn't so subtle. Any conceptual connection is at the level of the mathematical analogy you've already noticed. But as for a physical connection, the burden is on you to do a bit of legwork on what physics? Option pricing certainly doesn't result from the thermal jiggling of atoms. So why do you expect any connection? – wsc Mar 22 '12 at 20:15
wsc: There is indeed a juggling connection which is the underlying Brownian motion in real options theory. – S_H Mar 22 '12 at 20:32

As a curious fact, I was reading not long time ago the book When Genius Failed: The Rise and Fall of Long-Term Capital Management. There it is said that Myron Scholes and specially Robert C. Merton developped all their option pricing theory mimicking physical models [taken from research papers and books of statistical mechanics] and having faith in the so-called "efficient market hypothesis". The history of LTCM is well-known in finance, they lose billions of dollars after having been leveraged $250$ to $1$ [that is they invested $250$ dollars while having actually $1$].