The Feynman Kac Formula and Its Use in Pricing

The Feynman Kac formula proves that solving a pricing partial differential equation and taking a discounted expectation give the same price.

The Feynman-Kac formula connects a certain class of partial differential equations to expectations of functionals of a stochastic process, and in pricing theory it is the exact bridge that lets a derivative's price be computed either by solving a partial differential equation or by taking a discounted expectation under the risk neutral measure. The two routes must agree, and the formula is the theorem that proves they will.

Consider a function u of time and the state variable x that solves a partial differential equation of the form: the partial derivative with respect to time, plus mu times the first spatial partial, plus one half sigma squared times the second spatial partial, minus r times u, equals zero, with a terminal condition u at maturity equal to some payoff function of the terminal state. The Feynman-Kac formula states that the solution u at time t and state x equals the discounted expectation, under the probability measure for which the state process has drift mu and volatility sigma, of the payoff function evaluated at the terminal state, discounted at rate r from maturity back to time t.

The formula turns a boundary value problem in partial differential equations into an expectation over paths of a diffusion process, and the equivalence works both ways. Anyone who can solve the differential equation has the expectation for free, and anyone who can simulate or otherwise compute the expectation has the solution to the differential equation for free.

The proof applies Ito's lemma to the process u of t and X sub t, where X is the underlying diffusion. Because u solves the partial differential equation, the drift term in the resulting expansion collapses to r times u, which means the discounted process, defined as e to the minus r times t, times u of t and X sub t, has zero drift and is therefore a local martingale. Taking expectations at time t and at maturity, and using the terminal condition that u at maturity equals the payoff, gives exactly the discounted expectation formula stated above. The Ito expansion that makes this collapse work is covered step by step in Ito's Lemma and Stochastic Calculus Derivation.

The martingale property invoked partway through this argument is not incidental, and understanding why the discounted price process must have this property is what allows the drift terms to cancel so cleanly.