Feynman-Kac Formula

Motivation: why this matters in quant finance

Option pricing has two faces. The probabilistic face says "discount the expected payoff under the right dynamics"; the analytic face says "solve a parabolic PDE with a terminal payoff." The Feynman-Kac formula is the theorem that makes those faces the same object.

Lawler introduces it with a stock following geometric Brownian motion and a European call payoff

F(x)=(x-S)^+

. If

\phi(t,x)=\mathbb{E}\left[e^{-r(T-t)}F(X_T)\mid X_t=x\right],

then $\phi$ satisfies a PDE. In finance, this is why a Monte Carlo estimator and a finite-difference solver can agree: they are approximating the same conditional expectation from different sides.

The formula does not choose the pricing measure and does not prove absence of arbitrage. It starts after the diffusion dynamics and discounting convention have been specified, then turns the expectation into the corresponding differential equation.

The informal idea

Imagine holding a future payoff $F(X_T)$ when the state process currently has value $X_t=x$ . If a dollar at time $s$ grows at instantaneous rate $r(s,X_s)$ , then the payoff must be discounted back along the path. The value function is therefore

\phi(t,x)=\mathbb{E}\left[\exp\left(-\int_t^T r(s,X_s)\,ds\right)F(X_T)\mid X_t=x\right].

The key martingale is the discounted value process. Lawler writes it as

R_t^{-1}\phi(t,X_t)

, where

R_t

is the accumulated money-market account. Since it is a conditional expectation of the terminal discounted payoff, it must be a martingale. Applying Itô's lemma and forcing its

dt

term to vanish produces the PDE.

The whole proof idea is one sentence: make the discounted conditional expectation a martingale, then read off the zero-drift condition.

Formal definitions

Let $X_t$ solve the one-dimensional diffusion SDE

dX_t=m(t,X_t)\,dt+\sigma(t,X_t)\,dW_t.

Let $F(X_T)$ be an integrable payoff at time $T$ , and let $r(t,x)\ge0$ be a discounting rate. Define

\phi(t,x)=\mathbb{E}\left[\exp\left(-\int_t^T r(s,X_s)\,ds\right)F(X_T)\mid X_t=x\right].

If $\phi$ is $C^1$ in $t$ and $C^2$ in $x$ , Lawler's Feynman-Kac theorem gives

\partial_t\phi(t,x) =-m(t,x)\partial_x\phi(t,x) -\frac{1}{2}\sigma^2(t,x)\partial_{xx}\phi(t,x) +r(t,x)\phi(t,x),

for $0\le t<T$ , with terminal condition

\phi(T,x)=F(x).

Equivalently, using the time- $t$ generator

L_t f(x)=m(t,x)f'(x)+\frac{1}{2}\sigma^2(t,x)f''(x),

the PDE is

\partial_t\phi(t,x)+L_t\phi(t,x)-r(t,x)\phi(t,x)=0.

Key properties

It turns a discounted expectation into a PDE

The formula starts with a path expectation and proves that the value function satisfies a terminal-value PDE. Finance consequence: pricing by simulation and pricing by PDE are not rival models; they are two numerical routes to the same value under the same diffusion assumptions.

The discount rate appears as a killing term

In the generator form,

\partial_t\phi+L_t\phi-r\phi=0.

The $-r\phi$ term is exactly the mathematical trace of discounting. If $r=0$ , the equation reduces to the backward Kolmogorov equation for an undiscounted conditional expectation.

The terminal condition is the payoff

The PDE runs backward from

\phi(T,x)=F(x)

. In a European option, the payoff is known at expiry, while today's value is the unknown. Finance consequence: option PDEs are terminal-value problems, not initial-value problems.

The martingale argument depends on Markov state sufficiency

Lawler uses that, given $\mathcal{F}_t$ , the only relevant state information for predicting $X_T$ is $X_t$ . This is why the value can be written as $\phi(t,X_t)$ rather than as a functional of the whole past path. Path-dependent payoffs require enlarging the state before this formula applies directly.

Smoothness is an assumption in the introductory theorem

Lawler explicitly assumes $\phi$ is $C^1$ in time and $C^2$ in space, and notes that the general regularity theory is beyond the book. Kinked payoffs such as calls are handled by approximation or viscosity/weak-solution theory in more advanced treatments.

Worked examples

Example 1: no discounting recovers the backward equation

Set $r=0$ and define

u(t,x)=\mathbb{E}[F(X_T)\mid X_t=x].

Feynman-Kac becomes

\partial_tu(t,x)+L_tu(t,x)=0,\qquad u(T,x)=F(x).

This is the backward Kolmogorov equation. Feynman-Kac is therefore not separate from generators; it is the discounted payoff version of the same operator calculus.

Example 2: Lawler's geometric Brownian motion call

Let

dX_t=mX_t\,dt+\sigma X_t\,dW_t

and let $F(x)=(x-S)^+$ be a call payoff with strike $S$ . With constant discount rate $r$ ,

\phi(t,x)=\mathbb{E}\left[e^{-r(T-t)}F(X_T)\mid X_t=x\right].

The generator is

Lf(x)=mx f'(x)+\frac{1}{2}\sigma^2x^2f''(x),

so Feynman-Kac gives

\partial_t\phi+m x\,\partial_x\phi+\frac{1}{2}\sigma^2x^2\partial_{xx}\phi-r\phi=0.

Lawler notes that this is a version of the Black-Scholes PDE. In risk-neutral pricing, $m$ is replaced by $r$ , but that substitution comes from financial no-arbitrage reasoning, not from Feynman-Kac itself.

Example 3: deterministic discounting as a multiplier

If $r(t)$ is deterministic, Lawler writes

\phi(t,x)=\exp\left(-\int_t^T r(s)\,ds\right)f(t,x),

where

f(t,x)=\mathbb{E}[F(X_T)\mid X_t=x].

Since $f$ satisfies $\partial_t f=-L_tf$ , differentiating the product gives

\partial_t\phi=r(t)\phi-L_t\phi.

This is the same PDE:

\partial_t\phi+L_t\phi-r(t)\phi=0.

The example makes the discounting role transparent: it shifts the backward equation by a value-proportional term.

Common confusions and pitfalls

"Feynman-Kac proves the risk-neutral measure exists." No. It connects a specified diffusion expectation with a PDE. The risk-neutral measure comes from no-arbitrage and change-of-measure arguments such as Girsanov's theorem.

"The discount term has the wrong sign." In generator form the PDE is $\phi_t+L_t\phi-r\phi=0$ . Solving for $\phi_t$ moves the generator terms to the other side and writes $+r\phi$ there. Both forms are the same equation.

"Feynman-Kac applies to any path-dependent payoff as written." The formula uses a Markov state $X_t$ . For path-dependent products, the state must be enlarged to include the relevant running average, maximum, accrued integral, or barrier status.

"The PDE route avoids probabilistic assumptions." The PDE coefficients are exactly the diffusion coefficients. Changing $m$ , $\sigma$ , or the measure changes the PDE.

"A kinked payoff violates the theorem, so option pricing fails." The introductory theorem assumes smoothness of $\phi$ , not necessarily smoothness of $F$ in the final applied solution. Advanced regularity theory or approximation justifies common payoff cases.

Where this goes next

Black-Scholes PDE: applies Feynman-Kac to risk-neutral geometric Brownian motion.
Girsanov's Theorem: supplies the drift change needed before the pricing expectation is risk-neutral.
Infinitesimal Generators and Kolmogorov Equations: isolates the operator $L_t$ used in the PDE.
Risk-Neutral Measure: explains the finance assumption behind the expectation measure.
Local Martingales and Semimartingales: clarifies why the discounted value process must be a martingale, not merely formal notation.

References

Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 4 §4.3 (Feynman-Kac formula), with generator notation from Ch. 3 §3.5 (Diffusions).