In the book "The Poincare Conjecture: In Search of the Shape of the Universe" by Donal O'Shea, the author states that, "The Ricci-flow equation Perelman wrote, a type of heat equation, is a distant relative of the Black-Scholes equation."
I won't be taking PDE's until this fall, so it may be completely obvious to someone who has taken the course. But it's not clear just looking at the two equations why they are distant relatives.
Well, they are both famous PDEs :-)
The Ricci flow is a parabolic equation, and if you make an appropriate change of variables, so is Black-Scholes. So, speaking extremely roughly, they both have a structure that causes them to tend to "flatten out curves" in initial data.
Beyond that, I am not aware of any deep similarities or connections. It is possible the author just wanted to mention both because they sound cool.