# 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.

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You haven't accepted any answers to your previous questions. That red "0% accept rate" is going to put a lot of people off from helping you. –  chrisaycock Mar 22 '12 at 14:07
Oh ok. I am sorry about that. Thanks for informing me about the general norm. –  S_H Mar 22 '12 at 16:45
–  chrisaycock Mar 22 '12 at 16:50
Just accepted answers from all of my previous questions. –  S_H Mar 22 '12 at 18:21
Excellent. Thanks. –  chrisaycock Mar 22 '12 at 19:50

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.