Skip to content
Stochastic Calculus / Diffusions, SDEs, and the Feynman-Kac Formula

The Feynman-Kac Formula

The Black-Scholes PDE

∂tV+rS∂SV+12σ2S2∂SSV−rV=0,V(T,S)=(S−K)+\partial_t V + rS \partial_S V + \tfrac{1}{2}\sigma^2 S^2 \partial_{SS} V - rV = 0, \quad V(T, S) = (S-K)^+
is not an ad hoc PDE. It is the Feynman-Kac formula applied to the log-normal diffusion. The Feynman-Kac theorem says: whenever a random variable φ(t,x)\varphi(t, x) is defined as the conditional expectation E[e−∫tTr dsF(XT)∣Xt=x]\mathbb{E}[e^{-\int_t^T r \, ds} F(X_T) \mid X_t = x] along a diffusion dXt=m(t,Xt) dt+σ(t,Xt) dWtdX_t = m(t, X_t) \, dt + \sigma(t, X_t) \, dW_t, the function φ\varphi solves a parabolic PDE — and conversely, the solution to that PDE is that conditional expectation.

This duality is the bridge between stochastic analysis and PDEs. In finance it means two things: (1) you can price any European derivative as a PDE boundary value problem, and (2) Monte Carlo simulation (sampling paths of XtX_t and averaging F(XT)F(X_T)) is an alternative solver for those PDEs. The Feynman-Kac formula is why Black, Scholes, and Merton's "no-arbitrage PDE" and "risk-neutral expectation formula" are the same answer.

Learning objectives. By the end of this note you should be able to:
  • State the Feynman-Kac theorem and identify each component.
  • Derive the PDE from the conditional expectation representation using Itô's product rule.
  • Apply the theorem to recover the Black-Scholes PDE.
  • State the terminal condition and explain why it encodes the payoff.
  • Explain the PDE ↔ Monte Carlo duality.

In this lesson

  1. The Setting
  2. The Feynman-Kac Theorem
  3. Derivation via Itô's Product Rule
  4. The Generator Formulation
  5. Applications in Finance

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or explore the curriculum.