Feynman-Kac formula
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
Probabilistic representation of solutions to parabolic PDEs. For V(t,x) solving partial_t V + μ(x)partial_x V + 1/2σ^2(x)partial_xx V - rV = 0 with terminal condition V(T,x) = h(x), the solution is V(t,x) = mathbbE!left[e^-r(T-t),h(X_T),big|,X_t = xright], where X follows dX = μ(X)dt + σ(X)dW. The bridge between the BSM PDE and the risk-neutral expectation pricing formula — and the theoretical basis of Monte Carlo pricing.
Feynman-Kac formula in plain English
The Feynman-Kac formula is a beautiful bridge between two ways of solving the same problem: a hard calculus equation (a PDE) describing how a value spreads and changes, and a much friendlier idea — just simulate the random future many times and take the average. It says the answer to that gnarly equation equals the expected (average) payoff of a randomly wandering quantity, discounted back to today's value. Imagine you want to know a fair price for a bet whose outcome depends on a stock that jiggles randomly: instead of solving scary equations, you can 'roll the dice' thousands of times on possible stock paths, average the payoffs, and shrink that back for the passage of time. This is exactly why Monte Carlo simulation works for pricing options — it's the practical payoff of a deep theoretical result.
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