Chapter 4 Feynman-Kac and PDEs
4.1 The Bridge Between Probability and Analysis
One of the most beautiful results in mathematics connects two seemingly different worlds:
- Stochastic differential equations (SDEs) - the probabilistic world
- Partial differential equations (PDEs) - the analytical world
This connection, known as the Feynman-Kac formula, allows us to: - Solve PDEs by simulating random processes (Monte Carlo) - Solve SDEs by solving PDEs (numerical PDE methods) - Gain intuition about both simultaneously
4.2 The Backward Kolmogorov Equation
Consider an SDE:
\[dX(t) = \mu(X) \, dt + \sigma(X) \, dB(t)\]
Define the function:
\[u(t, x) = E[g(X(T)) \mid X(t) = x]\]
This represents the expected value of some payoff function \(g\) at terminal time \(T\), given that the process is at position \(x\) at time \(t\).
Question: What PDE does \(u(t,x)\) satisfy?
4.2.1 Heuristic Derivation
Apply Itô’s lemma to \(u(t, X(t))\):
\[du = \frac{\partial u}{\partial t}dt + \frac{\partial u}{\partial x}dX + \frac{1}{2}\frac{\partial^2 u}{\partial x^2}(dX)^2\]
Substitute \(dX = \mu \, dt + \sigma \, dB\) and \((dX)^2 = \sigma^2 \, dt\):
\[du = \left[\frac{\partial u}{\partial t} + \mu\frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2 u}{\partial x^2}\right]dt + \sigma\frac{\partial u}{\partial x}dB\]
Integrate from \(t\) to \(T\) and take expectations. The stochastic integral has zero expectation, so:
\[E[u(T, X(T))] - u(t, x) = E\left[\int_t^T \left(\frac{\partial u}{\partial s} + \mu\frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2 u}{\partial x^2}\right)ds\right]\]
At terminal time: \(u(T, X(T)) = g(X(T))\) by definition.
For this to hold for all paths, we need:
\[\boxed{\frac{\partial u}{\partial t} + \mu(x)\frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2(x)\frac{\partial^2 u}{\partial x^2} = 0}\]
with boundary condition \(u(T, x) = g(x)\).
This is the backward Kolmogorov equation (also called the Fokker-Planck backward equation).
4.3 Example: The Heat Equation
Take the simplest SDE: \(dX = dB\) (pure Brownian motion, \(\mu = 0\), \(\sigma = 1\)).
The backward equation becomes:
\[\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial^2 u}{\partial x^2} = 0\]
Change time direction by setting \(\tau = T - t\):
\[\boxed{\frac{\partial u}{\partial \tau} = \frac{1}{2}\frac{\partial^2 u}{\partial x^2}}\]
This is the heat equation!
4.3.1 Probabilistic Interpretation
The solution to the heat equation with initial condition \(u(0, x) = g(x)\) is:
\[u(\tau, x) = E[g(X(\tau)) \mid X(0) = x] = \int_{-\infty}^{\infty} g(y) \frac{1}{\sqrt{2\pi\tau}} e^{-(y-x)^2/(2\tau)} dy\]
The heat kernel \(\frac{1}{\sqrt{2\pi\tau}} e^{-(y-x)^2/(2\tau)}\) is precisely the transition density of Brownian motion!
Deep insight: Heat diffusion is mathematically identical to Brownian motion. Temperature spreads like a random walk.
4.4 The Black-Scholes PDE
Now consider geometric Brownian motion:
\[dS = \mu S \, dt + \sigma S \, dB\]
For a derivative with payoff \(g(S(T))\) at time \(T\), we want the price:
\[V(t, S) = E[g(S(T)) \mid S(t) = S]\]
But we need to account for discounting at the risk-free rate \(r\). In a risk-neutral world (more on this later), we replace \(\mu\) with \(r\) and define:
\[V(t, S) = e^{-r(T-t)}E^{\mathbb{Q}}[g(S(T)) \mid S(t) = S]\]
Let \(u(t, S) = e^{-r(T-t)}V(t, S)\) to factor out the discounting. By Itô’s lemma and similar arguments:
\[\boxed{\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} = rV}\]
with terminal condition \(V(T, S) = g(S)\).
This is the Black-Scholes PDE!
4.4.1 Solving for a European Call
For \(g(S) = \max(S - K, 0)\), the solution is:
\[V(t, S) = S\Phi(d_1) - Ke^{-r(T-t)}\Phi(d_2)\]
where: \[d_1 = \frac{\log(S/K) + (r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}, \quad d_2 = d_1 - \sigma\sqrt{T-t}\]
We’ll derive this in the next chapter using risk-neutral pricing.
4.5 The Forward Kolmogorov Equation
There’s also a forward equation that describes how the probability density \(p(t, x)\) of \(X(t)\) evolves:
\[\frac{\partial p}{\partial t} = -\frac{\partial}{\partial x}[\mu(x)p] + \frac{1}{2}\frac{\partial^2}{\partial x^2}[\sigma^2(x)p]\]
This is the forward Kolmogorov equation (or Fokker-Planck forward equation).
4.6 Feynman-Kac Formula (General Version)
The most general form handles running costs and discounting:
Theorem: Consider the SDE \(dX = \mu(X) \, dt + \sigma(X) \, dB\) with initial condition \(X(t) = x\).
Define:
\[u(t, x) = E\left[g(X(T))e^{-\int_t^T c(X(s))ds} + \int_t^T f(X(s))e^{-\int_t^s c(X(r))dr}ds \mid X(t) = x\right]\]
Then \(u\) satisfies:
\[\frac{\partial u}{\partial t} + \mu\frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2 u}{\partial x^2} - c(x)u + f(x) = 0\]
with \(u(T, x) = g(x)\).
Here: - \(c(x)\) is a discount rate (e.g., interest rate) - \(f(x)\) is a running payoff (e.g., dividends) - \(g(x)\) is terminal payoff
4.7 Applications
4.7.1 Option Pricing
The Feynman-Kac formula underpins all of option pricing theory: - Price = expected discounted payoff under risk-neutral measure - Can solve either the PDE or simulate the SDE
4.7.2 Statistical Mechanics
In physics, the heat equation describes: - Temperature diffusion - Particle density evolution - Quantum mechanics (Schrödinger equation with imaginary time)
4.8 Numerical Methods
4.8.1 Monte Carlo Simulation
To compute \(u(t,x) = E[g(X(T)) \mid X(t) = x]\):
- Simulate many paths of the SDE starting from \(X(t) = x\)
- Evaluate \(g(X(T))\) for each path
- Average the results
Advantages: Easy to implement, works in high dimensions Disadvantages: Slow convergence (\(\mathcal{O}(1/\sqrt{N})\)), difficult for early exercise
4.9 Summary
The Feynman-Kac connection is profound:
SDE: dX = μdt + σdB
↓ (Itô's lemma)
PDE: ∂u/∂t + μ∂u/∂x + ½σ²∂²u/∂x² = 0
↕ (Feynman-Kac)
Probabilistic representation: u(t,x) = E[g(X(T)) | X(t)=x]This bridges: - Probability ↔︎ Analysis - Random walks ↔︎ Differential equations - Simulation ↔︎ Analytic solutions - Physics ↔︎ Finance
It’s one of the most beautiful and useful results in applied mathematics.