# 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.

## 2 Answers

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.

• In physics we have an initial condition, such as the initial distribution of particles in a system and then we measure quantities related to the system as time flows forward. In finance we have a terminal condition, the payoff and we calculate quantities related to the payoff (value, delta, etc.) at earlier times. Unfortunately diffusion equations do not have the time reversal symmetry of say Newton's equations, so we can't just say, let's pretend that time flows backward and make an analogy between physics and finance. That was my point. – mepuzza Mar 23 '12 at 9:10

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.