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 $\mu_i = r x_i$, $a_{ij} = \rho_{ij}\sigma_i\sigma_j x_i x_j$.

Limitations

Regularity requirement. Feynman-Kac in its classical form requires $u \in C^{1,2}$. For discontinuous payoffs (barrier options, digital options), the terminal condition is discontinuous and the classical $C^{1,2}$ solution may fail to exist. The correct framework is viscosity solutions (Crandall-Lions), which extends Feynman-Kac to non-smooth settings.

Markov property. The probabilistic representation $u(t, x) = \mathbb{E}[\cdot \mid X_t = x]$ requires the underlying process to be Markovian — the future distribution depends only on the current state, not the path history. For non-Markovian models (path-dependent volatility, rough volatility with memory), Feynman-Kac does not apply in its classical form. The correct extension uses backward SDEs (BSDEs): the value process $(Y_t, Z_t)$ satisfies $dY_t = -f(t, Y_t, Z_t) dt + Z_t \, dW_t$, $Y_T = \varphi(X_T)$.

Uniqueness. The Feynman-Kac formula identifies a particular probabilistic solution of the PDE. Without growth conditions at infinity (e.g., $|u(t,x)| \leq C(1 + |x|^p)$), the PDE may admit multiple solutions. The financial interpretation selects the relevant one: the smallest non-negative solution for put prices, for instance.

American options. For American options, the option holder has the right to exercise at any stopping time $\tau \leq T$. The value satisfies a variational inequality (free-boundary problem): $\min\!\left(-\frac{\partial V}{\partial t} - \mathcal{L}V + rV,\ V - (s - K)^+\right) = 0.$ In the continuation region (where $V > (s - K)^+$), the PDE holds. At the exercise boundary, the value equals the intrinsic value. PSOR solves this numerically.

Interview Angle

L1: State the Black-Scholes PDE and its terminal and boundary conditions for a European call. What is the economic interpretation of each term?

Black-Scholes PDE and conditions. $\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV = 0, \quad (t, S) \in [0,T) \times (0,\infty).$ Terminal: $V(T, S) = (S - K)^+$. Boundary: $V(t, 0) = 0$; $V(t, S) \sim S - Ke^{-r(T-t)}$ as $S \to \infty$.

Economic interpretation of each term. Consider a delta-hedged portfolio $\Pi = V - \Delta S$, held for time $dt$:

• $\partial V/\partial t$: Theta — the time decay of the option value. For a long call, $\Theta < 0$: the option loses value as expiry approaches, all else equal.
• $rS \partial V/\partial S$: The drift of $S$ under $\mathbb{Q}$ is $rS \, dt$. This contributes $r S \Delta \, dt$ to the portfolio's change in value — the "rho from the drift" term.
• $\frac{1}{2}\sigma^2 S^2 \partial^2 V/\partial S^2$: Gamma P&L — the convexity income. A delta-hedged long option position benefits from large moves in either direction. When $S$ moves by $dS$, the hedging error is $\frac{1}{2}\Gamma(dS)^2 = \frac{1}{2}\Gamma\sigma^2 S^2 dt$. This is the compensation the option buyer receives for holding positive convexity.
• $-rV$: Financing cost — funding the option position at rate $r$ costs $rV \, dt$. Equivalently: the option premium invested at $r$ grows at $rV$, which must be subtracted.

The PDE states that in continuous time, the gamma income exactly offsets the theta decay plus the net financing cost of the delta hedge. This is the no-arbitrage condition: a self-financing delta-hedged portfolio earns zero riskless profit.

L2: Derive the Black-Scholes PDE using Feynman-Kac (not delta hedging). What regularity conditions on $V$ are required? What does the Markov property of GBM provide?

Derivation via Feynman-Kac. Under the risk-neutral measure $\mathbb{Q}$, the stock satisfies $dS_s = rS_s \, ds + \sigma S_s \, d\widetilde{W}_s$. The no-arbitrage price of a European payoff $\varphi(S_T)$ is: $V(t, S_t) = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}[\varphi(S_T) \mid S_t].$ Define $u(t, x) = e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[\varphi(S_T) \mid S_t = x]$. This matches the Feynman-Kac setup with $\mu(s,x) = rx$, $\sigma(s,x) = \sigma x$, constant discount rate $r$, and terminal condition $u(T,x) = \varphi(x)$. The Feynman-Kac theorem then states that $u$ satisfies: $\frac{\partial u}{\partial t} + rx\frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2 x^2 \frac{\partial^2 u}{\partial x^2} - ru = 0,$ which is precisely the Black-Scholes PDE with $x = S$. The derivation reduces entirely to the Feynman-Kac theorem applied to the GBM generator; no self-financing portfolio or hedging argument is needed.

Regularity requirement. The classical Feynman-Kac theorem requires $u \in C^{1,2}([0,T) \times (0,\infty))$ — once continuously differentiable in $t$, twice in $S$. For a European call, $\varphi(x) = (x-K)^+$ has a kink at $x = K$: the terminal condition is not $C^2$. The PDE regularity $u \in C^{1,2}$ holds for all $t < T$ (the heat kernel smooths the terminal kink for any $t < T$), but fails at $t = T$ at $S = K$. Practically, one applies Feynman-Kac on $[0, T-\varepsilon]$ and takes $\varepsilon \to 0$, or appeals to weak/viscosity solution theory to handle the terminal kink directly.

Markov property. GBM is a Markov process: the conditional distribution of $(S_s)_{s > t}$ given $\mathcal{F}_t$ depends only on $S_t$, not on the path history $(S_u)_{u \leq t}$. This means: $\mathbb{E}^{\mathbb{Q}}[\varphi(S_T) \mid \mathcal{F}_t] = \mathbb{E}^{\mathbb{Q}}[\varphi(S_T) \mid S_t] = u(t, S_t),$ i.e., the conditional expectation is a function of the current state $(t, S_t)$ alone. Without the Markov property, the pricing function would depend on the entire path history and could not be expressed as a finite-dimensional PDE. For example, an Asian option depends on the path average $\bar{S}_t = \frac{1}{t}\int_0^t S_s \, ds$; the state space must be augmented to $(t, S_t, \bar{S}_t)$ to recover the Markov property, and the resulting Feynman-Kac PDE is two-dimensional in the spatial variables.

L3: How does Feynman-Kac extend to American options? What replaces the PDE in the exercise region? Why does the classical Feynman-Kac fail for rough volatility models, and what is the correct replacement?

American options: optimal stopping and the free-boundary problem. An American option gives the holder the right to exercise at any $\mathbb{F}$-stopping time $\tau \leq T$. The value is: $V(t, S_t) = \sup_{\tau \in \mathcal{T}_{t,T}} \mathbb{E}^{\mathbb{Q}}\!\left[e^{-r(\tau - t)} g(S_\tau) \mid S_t\right],$ where $g(S) = (K - S)^+$ for an American put. The state space $(t, S)$ splits into two regions:

• Continuation region $\mathcal{C} = \{(t,S) : V(t,S) > g(S)\}$: it is not yet optimal to exercise. Here the option price satisfies the Black-Scholes PDE: $V_t + rS V_S + \tfrac{1}{2}\sigma^2 S^2 V_{SS} - rV = 0.$
• Exercise (stopping) region $\mathcal{E} = \{(t,S) : V(t,S) = g(S)\}$: exercise immediately. Here $V = g$ and the PDE is replaced by the inequality $-V_t - rS V_S - \tfrac{1}{2}\sigma^2 S^2 V_{SS} + rV \geq 0$ (the option has intrinsic value at least equal to holding costs).

The full system is the variational inequality: $\min\!\left(-V_t - rS V_S - \tfrac{1}{2}\sigma^2 S^2 V_{SS} + rV,\; V - g\right) = 0, \quad V(T,S) = g(S).$ The exercise boundary $S^*(t)$ separating $\mathcal{C}$ from $\mathcal{E}$ is the free boundary — it is part of the solution, not a given. At $S^*(t)$: value matching $V(t, S^*) = g(S^*)$ and smooth pasting $V_S(t, S^*) = g'(S^*)$ (the delta of the option equals the delta of the payoff at the boundary, eliminating a kink that would otherwise create an arbitrage). Numerically, PSOR (projected successive over-relaxation) solves this by projecting the finite-difference iterate onto the constraint $V \geq g$ at each grid point.

Classical Feynman-Kac does not apply here because the optimal stopping problem is not a fixed terminal condition problem: the effective "terminal" condition depends on where the boundary is, which is itself unknown.

Rough volatility and the failure of classical Feynman-Kac. In the rough Bergomi or rough Heston model, the instantaneous variance $\nu_t$ is driven by fractional Brownian motion with Hurst exponent $H \approx 0.1$: $\log\nu_t = \log\xi_0(t) + \eta\left(2H\right)^{1/2}\int_{-\infty}^t (t-s)^{H-1/2} \, dW_s^{\mathbb{Q}}.$

Two structural failures prevent Feynman-Kac from applying:

1. Lack of Markov property. Fractional BM has correlated increments: the future increment $W^H_{t+h} - W^H_t$ is correlated with the entire history $(W^H_s)_{s \leq t}$. The future distribution of $\nu$ depends on the full path $(\nu_s)_{s \leq t}$, not just $\nu_t$. The option price $V = V(t, \omega)$ is a functional of the path, not a function of a finite state. There is no finite-dimensional Feynman-Kac PDE.

2. Non-semimartingale structure. For $H \neq 1/2$, fBM is not a semimartingale: it cannot be decomposed as a local martingale plus a finite-variation process. Itô's lemma and the generator $\mathcal{L}$ are undefined for non-semimartingales. The entire PDE derivation in the Feynman-Kac proof — which crucially uses Itô's lemma to compute $d(e^{-\int r \, ds} u(s, X_s))$ — breaks down.

Correct replacement: BSDEs. The natural extension in non-Markovian settings is the backward SDE (BSDE). One seeks adapted processes $(Y_t, Z_t)$ satisfying: $dY_t = -f(t, Y_t, Z_t) \, dt + Z_t \, dW_t, \quad Y_T = \xi,$ where $\xi = \varphi(S_T)$ is the terminal payoff and $f$ is a driver encoding the discount rate and hedging costs. Under Lipschitz conditions on $f$ (Pardoux-Peng, 1990), the BSDE admits a unique adapted solution. The process $Y_t$ represents the option price, and $Z_t$ the hedging ratio — without requiring the state to be finite-dimensional or the filtration to be Markovian. For rough volatility specifically, the practical approach is Monte Carlo simulation (e.g., the hybrid scheme of Bennedsen–Lunde–Pakkanen for discretising the fBM convolution) combined with regression-based methods (least-squares Monte Carlo, neural-network pricing) to compute conditional expectations in the non-Markovian setting.