Feynman-Kac and the Connection to PDEs

Setup

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space with filtration $\mathbb{F} = (\mathcal{F}_t)_{t \geq 0}$. Consider the SDE: $dX_s = \mu(s, X_s) \, ds + \sigma(s, X_s) \, dW_s, \qquad X_t = x,$ started at time $t$ from position $x \in \mathbb{R}$. We assume $\mu$ and $\sigma$ are globally Lipschitz and have at most linear growth in $x$ — conditions ensuring strong existence and pathwise uniqueness of the solution $X_s^{t,x}$ for $s \geq t$.

The generator of the diffusion is the second-order differential operator: $\mathcal{L} = \mu(t, x)\frac{\partial}{\partial x} + \frac{1}{2}\sigma^2(t, x)\frac{\partial^2}{\partial x^2}.$

Feynman-Kac Theorem

Theorem. Let $r: [0,T] \times \mathbb{R} \to \mathbb{R}_+$ be a bounded discount function, and $\varphi: \mathbb{R} \to \mathbb{R}$ a measurable terminal payoff with $\mathbb{E}[|\varphi(X_T^{t,x})|] < \infty$. Define: $u(t, x) = \mathbb{E}\!\left[\exp\!\left(-\int_t^T r(s, X_s) \, ds\right) \varphi(X_T) \,\Bigg|\, X_t = x\right].$

If $u \in C^{1,2}([0,T) \times \mathbb{R})$, then $u$ is the unique solution to the backward parabolic PDE: $\frac{\partial u}{\partial t} + \mathcal{L}u - r(t, x) \, u = 0, \qquad (t, x) \in [0, T) \times \mathbb{R},$ $u(T, x) = \varphi(x).$

Equivalently: $\frac{\partial u}{\partial t} + \mu(t,x)\frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2(t,x)\frac{\partial^2 u}{\partial x^2} - r(t,x) \, u = 0.$

The PDE is solved backward in time from the terminal condition $u(T, x) = \varphi(x)$.

Derivation

Define the discounted value process: $Y_s = e^{-\int_t^s r(u, X_u) \, du} \, u(s, X_s), \qquad s \in [t, T].$

We claim $Y_s$ is a local martingale if and only if $u$ satisfies the PDE.

Apply Itô's lemma to $u(s, X_s)$: $du(s, X_s) = \left(u_t + \mu u_x + \frac{1}{2}\sigma^2 u_{xx}\right) ds + \sigma u_x \, dW_s.$

Let $D_s = e^{-\int_t^s r \, du}$ denote the discount factor. Then $dD_s = -r(s, X_s) D_s \, ds$. Apply the product rule $d(D_s u_s) = D_s \, du_s + u_s \, dD_s + d\langle D, u \rangle_s$. Since $D_s$ has finite variation, the covariation term vanishes: $dY_s = D_s\left(u_t + \mu u_x + \frac{1}{2}\sigma^2 u_{xx} - r \, u\right) ds + D_s \sigma u_x \, dW_s.$

For $Y_s$ to be a local martingale, the $ds$ term must vanish identically. This forces: $u_t + \mu u_x + \frac{1}{2}\sigma^2 u_{xx} - r \, u = 0.$

This is the Feynman-Kac PDE. Taking expectations at $s = T$: $\mathbb{E}[Y_T \mid \mathcal{F}_t] = Y_t \quad \Rightarrow \quad u(t, x) = \mathbb{E}\!\left[e^{-\int_t^T r \, ds} \varphi(X_T) \mid X_t = x\right]. \qquad \square$

The Black-Scholes PDE

Under the risk-neutral measure $\mathbb{Q}$, the stock follows: $dS_s = r S_s \, ds + \sigma S_s \, d\widetilde{W}_s.$

The risk-neutral pricing formula for a European derivative with payoff $\varphi(S_T)$: $V(t, S_t) = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}[\varphi(S_T) \mid S_t].$

Matching with Feynman-Kac: $X_s = S_s$, $\mu(s, x) = rx$, $\sigma(s, x) = \sigma x$, constant discount rate $r$. The function $V(t, s)$ satisfies: $\boxed{\frac{\partial V}{\partial t} + rs\frac{\partial V}{\partial s} + \frac{1}{2}\sigma^2 s^2 \frac{\partial^2 V}{\partial s^2} - rV = 0,}$ with terminal condition $V(T, s) = \varphi(s)$.

This is the Black-Scholes PDE, derived from no-arbitrage (via Girsanov) and Feynman-Kac — not from the original delta-hedging argument.

Boundary Conditions for a European Call

For $\varphi(s) = (s - K)^+$:

• Terminal: $V(T, s) = (s - K)^+$.
• At $s = 0$: $V(t, 0) = 0$. Under GBM, $S = 0$ is an absorbing barrier; a call on a worthless stock is worthless.
• As $s \to \infty$: $V(t, s) \sim s - K e^{-r(T-t)}$. A deep-in-the-money call behaves like a forward contract.

Feynman-Kac vs Delta Hedging

The original Black-Scholes (1973) derivation constructed a delta-hedged portfolio, assumed it earns the risk-free rate, and derived the PDE by no-arbitrage. Feynman-Kac provides a cleaner, more general route:

1. No-arbitrage $\Rightarrow$ EMM $\mathbb{Q}$ exists (FTAP).
2. Risk-neutral pricing: $V_t = e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[\varphi(S_T) \mid S_t]$.
3. Feynman-Kac: this expectation satisfies the BS PDE.

The delta-hedging argument assumes the hedge portfolio is self-financing and that the hedging error is zero — implicitly assuming completeness. Feynman-Kac makes the probabilistic structure transparent and extends immediately to:

• Multidimensional underlyings (rainbow options, basket options).
• Time-dependent rates and volatility.
• Path-dependent payoffs via the Markov property and a sufficient state variable extension.

Multi-Dimensional Feynman-Kac

Let $X = (X^1, \ldots, X^d)$ be a vector Itô process with generator: $\mathcal{L} = \sum_{i=1}^d \mu_i \frac{\partial}{\partial x_i} + \frac{1}{2}\sum_{i,j=1}^d a_{ij} \frac{\partial^2}{\partial x_i \partial x_j},$ where $a_{ij} = (\sigma\sigma^\top)_{ij}$ is the diffusion matrix. Feynman-Kac applies directly: the function $u(t, x) = \mathbb{E}\!\left[e^{-\int_t^T r \, ds} \varphi(X_T) \mid X_t = x\right]$ solves $\partial_t u + \mathcal{L}u - ru = 0$. The Black-Scholes PDE for a multi-asset option in a correlated log-normal model is the direct application of this with $\mu_i = r x_i$, $a_{ij} = \rho_{ij}\sigma_i\sigma_j x_i x_j$.