Feynman-Kac Theorem
Feynman-Kac Theorem states that the price of a derivative can be represented as the solution to a certain PDE. Specifically, if \(f(t, S)\) is the price of the derivative at time \(t\) when the underlying asset price is S which follows
\[dS(t) = rS(t)dt + σS(t)d \tilde{W}(t)\],then \(f(t,S)\) solves the following PDE:
\[\frac{\partial f}{\partial t} + rS \frac{\partial f}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 f}{\partial S^2} = r f\]with the terminal condition \(f(T, S) = h(S)\), where \(h(S)\) is the payoff at maturity \(T\). Document below provides some concrete examples on how Feynman-Kac Theorem can be used for pricing derivatives!