# 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. – Pupil 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. – Pupil Mar 22 '12 at 20:32

## 1 Answer

This was intended to be a comment, but is too long so I will post it as an answer. First of all, a disclaimer, I am a physict and all that I know about quantitative finance comes from self-learning, so please feel free of correcting me if I am mistaken (also in the physics stuff, of course!).

I have been doing a little of research, and perhaps you are right in your last point. The Black-Scholes PDE relies in the assumption of (i) the option prize is a continuous function of time and the undelying asset and (ii) the current stock price follows a [geometric] Brownian motion. From the physics point of view, there is a deep connection between Brownian motion and the diffusion equation which is exemplified in the famous Einstein relation. As the heat equation is a particular form of the diffusion equation, it is not so surprising that the heat kernel appears in some of the solutions of the Black-Scholes PDE.

However, there is no physical similarity between these two phenomena as they both do not describe physical processes but only are based in the same mathematics [stochastic calculus].

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 $1$].

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That helps, especially the last paragraph which I did not know earlier! Thanks. – Pupil Mar 22 '12 at 21:59
Can you provide some information as to where in the book is it mentioned that "Myron Scholes and specially Robert C. Merton developped all their option pricing theory mimicking physical models" – Pupil Mar 23 '12 at 2:35
On page 69 it says:"Of course, Merton's entire oeuvre depended on his assumptions about random walks, with their close tie to the physical world. As (Rosenfeld), he and his fellow Merton protégés used to run to the physics library looking for formulaic solutions that they could 'jam into finance'". I think that the link with statistical mechanics I made up myself, as it is the discipline where I know you use random walks... – DaniH Mar 23 '12 at 8:10
Then, on the next page, talking about Rosenfeld who had studied with Merton at Harvard says: "Kapor (his friend) was so intrigued by his friend's account of how stocks mimicked molecules that he enrolled at MIT..." – DaniH Mar 23 '12 at 8:13
Perhaps the interpretation I gave in the response on how Merton was based in physics to construct his option pricing model is a bit exagerated. I wonder if he was not guided by mathematics themselves instead. However, his basic assumptions, continuous time and brownian motion are rooted in statistical mechanics, this is for sure. – DaniH Mar 23 '12 at 8:16