The Feynman-Kac formula represents the solution of a parabolic PDE as an expectation over paths of a diffusion process. This connection allows you to:
ASolve any PDE exactly by computing a single stochastic integral
BEither solve PDEs by Monte Carlo simulation of the diffusion, or solve stochastic problems by PDE methods — the bridge works in both directions
CReplace all PDE theory with probability theory, since the stochastic representation is always more efficient
DOnly compute the solution at the boundary, not in the interior of the domain
The Feynman-Kac bridge works both ways. Given a PDE, you can estimate its solution by simulating many paths of the corresponding diffusion and averaging the payoff — this is the Monte Carlo method, which scales to high dimensions where grid-based PDE solvers fail. Conversely, given a stochastic expectation E[g(X(T))], you can recognize it as the solution of a PDE and apply analytical or numerical PDE techniques. Neither direction dominates — the choice depends on the dimension, regularity, and structure of the problem.
Question 2 Multiple Choice
In the Black-Scholes model, the option price V(S,t) satisfies the PDE ∂V/∂t + rS·∂V/∂S + (1/2)σ²S²·∂²V/∂S² - rV = 0. Via Feynman-Kac, this PDE is equivalent to:
AV(S,t) = E[e^{-r(T-t)} payoff(S(T)) | S(t) = S] under the risk-neutral measure, where dS = rS dt + σS dW̃
BV(S,t) = E[payoff(S(T)) | S(t) = S] under the physical measure, where dS = μS dt + σS dW
CV(S,t) = the probability that S(T) > K
DV(S,t) = e^{-rT}·payoff(S₀)
The Black-Scholes PDE is exactly in Feynman-Kac form with drift coefficient rS, diffusion σS, discount rate r, and terminal condition g(S) = payoff(S). The Feynman-Kac representation gives V(S,t) = E[e^{-r(T-t)}g(S(T)) | S(t) = S] where S follows dS = rS dt + σS dW̃. This is the risk-neutral pricing formula — the option price equals the discounted expected payoff under the risk-neutral measure. The connection between the PDE and the expectation is precisely the Feynman-Kac formula.
Question 3 Short Answer
Explain why the Feynman-Kac formula is especially useful in high-dimensional problems (e.g., options on multiple assets).
Think about your answer, then reveal below.
Model answer: Grid-based PDE solvers suffer from the curse of dimensionality: a grid with N points per dimension in d dimensions requires N^d total points, making computation intractable for d > 3-4. The Monte Carlo method from the Feynman-Kac representation simulates independent paths of the d-dimensional diffusion and averages the payoff. The convergence rate of Monte Carlo (1/√n for n paths) is independent of dimension — 10,000 paths give roughly the same accuracy whether d = 1 or d = 100. This dimension-independence makes the stochastic representation computationally superior for high-dimensional PDEs.
The Feynman-Kac formula transforms a d-dimensional PDE into an expectation over d-dimensional diffusion paths. While the PDE approach requires discretizing a d-dimensional domain, the Monte Carlo approach samples individual paths. Each path is one-dimensional (a sequence of time steps), regardless of d. This is why Monte Carlo pricing dominates in multi-asset derivatives and why the Feynman-Kac connection is practically important, not just theoretically elegant.