Hi, When I took an advanced partial differential equations course, the heat kernel part aroused great interest to me. Later when I read more relevent with heat kernel equations, I find that the heat kernel equations are in some way related to the Brownian Motion. Since the Browian motion is the theory foundation for the stochastic process and stochastic process is so important to finance thoery, can we try to figure out how to make the heat kernel equations to serve for the finance theory? Or is my thinking right?I need your advice!Thanks in advance!Mackinn

this is at the heart of the equivalence of the two approaches to derivatives pricing. The martingale approach relies on Brownian motion, the other relies on PDEs which are equivalent to the heat equation. The Feynman-Kac theorem is one way to go between the two. MJ

Thanks for your reply, MJ. You know I am a newcomer into finance. Please forgive my ignorance. I am quite confused of some definitions. You mean that there are two approaches to derivative pricing, martingale approach and finite difference. the first depends on B.M. and the latter relies in PDE. Right? Just now I checked the Feynman-Kac formula, i see that it is quite silimar to the heat kernel equation, but also with stochastic process. So that is the combination of the two principles. Are there any others? Mackinn

Well Brownian motion is fundamental in both approaches. In both cases one assumes the same stochastic process for the stock.In the PDE approach you deduce a PDE which the option price must satisfy for avoidance of arbitrage and then use PDE methods.In the martingale approach you show that the arbitrage-free price of an option is equal to a certain expectation which you then compute using probabilisitic methods. They give the same answers in the end.For PDEs approach see Wilmott's books.For expectations, see Baxter and Rennie.For connections between them see Bjork.MJ

Thank you very much, MJ. It seems more clearer to me. I will try to find the books to read and figure that out. Mackinn