Connection to Feynman-Kac¶

The Feynman-Kac theorem provides the bridge between probabilistic expectations and partial differential equations. For affine processes, this bridge has a remarkable consequence: the exponential-affine ansatz reduces the Feynman-Kac PDE to the Riccati ODE system, giving a complete and explicit connection between the probabilistic and analytical approaches to derivative pricing. This section develops the affine Feynman-Kac framework, derives the PDE, and shows how the ansatz produces the Riccati equations.

Learning Objectives

By the end of this section, you will be able to:

State the Feynman-Kac theorem for discounted expectations
Write the associated PDE for the discounted transform of an affine process
Reduce the PDE to the Riccati system using the exponential-affine ansatz
Verify the Feynman-Kac solution for the Black-Scholes model

Intuition¶

Derivative pricing has two equivalent formulations. The probabilistic approach computes \(V(t, x) = \mathbb{E}^{\mathbb{Q}}[e^{-\int_t^T r_s\,ds}\,h(X_T) \mid X_t = x]\) as a conditional expectation. The analytical approach solves the PDE \(\frac{\partial V}{\partial t} + \mathcal{A}V - rV = 0\) with terminal condition \(V(T, x) = h(x)\). The Feynman-Kac theorem says these are the same object.

For general processes, solving the PDE numerically is expensive---finite difference or finite element methods scale poorly with dimension. For affine processes, the exponential-affine ansatz transforms the PDE into ODEs, collapsing the computational cost from exponential in \(d\) to polynomial. This is why affine models dominate in multi-factor applications.

The Feynman-Kac Theorem¶

General Statement¶

Theorem (Feynman-Kac). Let \(X_t\) be an Ito diffusion on \(\mathbb{R}^d\) with generator

\[ \mathcal{A} = \sum_{i=1}^d b_i(x)\frac{\partial}{\partial x_i} + \frac{1}{2}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial^2}{\partial x_i \partial x_j} \]

Let \(c : \mathbb{R}^d \to \mathbb{R}\) be a continuous function (the "killing rate") and \(h : \mathbb{R}^d \to \mathbb{R}\) a measurable terminal condition. Define

\[ V(t, x) = \mathbb{E}\!\left[e^{-\int_t^T c(X_s)\,ds}\,h(X_T) \mid X_t = x\right] \]

Under suitable regularity conditions (Lipschitz coefficients, polynomial growth of \(h\)), \(V\) is the unique classical solution of the PDE:

\[ \frac{\partial V}{\partial t}(t, x) + \mathcal{A}V(t, x) - c(x)\,V(t, x) = 0, \qquad V(T, x) = h(x) \]

The term \(-c(x)V\) is the "killing" or "discounting" term. In finance, \(c(x) = r(x)\) is the short rate, and \(V\) is the time-\(t\) value of a claim paying \(h(X_T)\) at time \(T\).

Backward Formulation¶

Switching to time-to-maturity \(\tau = T - t\):

\[ \frac{\partial V}{\partial \tau}(\tau, x) = \mathcal{A}V(\tau, x) - c(x)\,V(\tau, x), \qquad V(0, x) = h(x) \]

This is the form most convenient for the Riccati derivation, since \(\tau\) increases from the terminal condition.

Affine Feynman-Kac PDE¶

Setup for Affine Processes¶

For an affine process on \(D = \mathbb{R}^m_+ \times \mathbb{R}^{d-m}\) with affine drift \(b(x) = b_0 + \sum_j b_j x_j\), affine diffusion \(a(x) = a_0 + \sum_j a_j x_j\), and affine killing rate \(c(x) = \rho_0 + \langle \rho_1, x \rangle\), the Feynman-Kac PDE becomes:

\[ \frac{\partial V}{\partial \tau} = \sum_{i=1}^d \left(b_{0,i} + \sum_{j=1}^d b_{j,i}\,x_j\right)\frac{\partial V}{\partial x_i} + \frac{1}{2}\sum_{i,k=1}^d \left(a_{0,ik} + \sum_{j=1}^d a_{j,ik}\,x_j\right)\frac{\partial^2 V}{\partial x_i \partial x_k} - (\rho_0 + \langle \rho_1, x \rangle)\,V \]

Exponential-Affine Terminal Condition¶

For the discounted transform, the terminal condition is \(h(x) = e^{\langle u, x \rangle}\). The PDE becomes:

\[ \frac{\partial V}{\partial \tau} = \mathcal{A}V - c(x)\,V, \qquad V(0, x) = e^{\langle u, x \rangle} \]

Reduction to the Riccati System¶

The Ansatz¶

Motivated by the terminal condition, substitute \(V(\tau, x) = \exp(\tilde{\phi}(\tau) + \langle \tilde{\psi}(\tau), x \rangle)\) with \(\tilde{\phi}(0) = 0\) and \(\tilde{\psi}(0) = u\).

Computing the derivatives:

\[ \frac{\partial V}{\partial \tau} = (\tilde{\phi}' + \langle \tilde{\psi}', x \rangle)\,V \]

\[ \frac{\partial V}{\partial x_i} = \tilde{\psi}_i\,V, \qquad \frac{\partial^2 V}{\partial x_i \partial x_k} = \tilde{\psi}_i\,\tilde{\psi}_k\,V \]

Substitution¶

Substituting into the PDE and dividing by \(V > 0\):

\[ \tilde{\phi}' + \langle \tilde{\psi}', x \rangle = \sum_i (b_{0,i} + \sum_j b_{j,i} x_j)\tilde{\psi}_i + \frac{1}{2}\sum_{i,k}(a_{0,ik} + \sum_j a_{j,ik} x_j)\tilde{\psi}_i\tilde{\psi}_k - \rho_0 - \langle \rho_1, x \rangle \]

Collecting Terms¶

Grouping by powers of \(x\):

Constant terms (\(x^0\)):

\[ \tilde{\phi}'(\tau) = \langle b_0, \tilde{\psi} \rangle + \frac{1}{2}\langle \tilde{\psi}, a_0\tilde{\psi} \rangle - \rho_0 = F(\tilde{\psi}) - \rho_0 \]

Linear terms (coefficient of \(x_j\)):

\[ \tilde{\psi}_j'(\tau) = \langle b_j, \tilde{\psi} \rangle + \frac{1}{2}\langle \tilde{\psi}, a_j\tilde{\psi} \rangle - \rho_{1,j} = R_j(\tilde{\psi}) - \rho_{1,j} \]

This recovers the extended Riccati system with discounting:

\[ \tilde{\phi}' = F(\tilde{\psi}) - \rho_0, \qquad \tilde{\psi}' = R(\tilde{\psi}) - \rho_1 \]

The Feynman-Kac PDE, which is a \((d+1)\)-dimensional partial differential equation, has been reduced to a \((d+1)\)-dimensional system of ordinary differential equations. For \(d = 1\), this reduces the PDE to two scalar ODEs; for \(d = 2\) (e.g., Heston), to three scalar ODEs.

Verification for the Black-Scholes Model¶

Setup¶

In the Black-Scholes model, \(X_t = \log S_t\) with \(dX_t = (r - \frac{1}{2}\sigma^2)\,dt + \sigma\,dW_t\) and \(c(x) = r\) (constant short rate). The affine parameters are \(b_0 = r - \frac{1}{2}\sigma^2\), \(b_1 = 0\), \(a_0 = \sigma^2\), \(a_1 = 0\), \(\rho_0 = r\), \(\rho_1 = 0\).

Riccati System¶

\[ \tilde{\psi}'(\tau) = 0, \qquad \tilde{\psi}(0) = u \]

\[ \tilde{\phi}'(\tau) = (r - \tfrac{1}{2}\sigma^2)\tilde{\psi} + \tfrac{1}{2}\sigma^2\tilde{\psi}^2 - r, \qquad \tilde{\phi}(0) = 0 \]

Solution¶

\(\tilde{\psi}(\tau) = u\) (constant), and:

\[ \tilde{\phi}(\tau) = \left[(r - \tfrac{1}{2}\sigma^2)u + \tfrac{1}{2}\sigma^2 u^2 - r\right]\tau \]

The Discounted Characteristic Function¶

Setting \(u = iv\):

\[ \tilde{\phi}(\tau, iv) = \left[(r - \tfrac{1}{2}\sigma^2)iv - \tfrac{1}{2}\sigma^2 v^2 - r\right]\tau \]

\[ \tilde{\psi}(\tau, iv) = iv \]

The discounted characteristic function is:

\[ \mathbb{E}\!\left[e^{-r\tau + iv\log S_T} \mid \log S_t = x\right] = \exp\!\left(\left[(r - \tfrac{1}{2}\sigma^2)iv - \tfrac{1}{2}\sigma^2 v^2 - r\right]\tau + ivx\right) \]

This is the standard Black-Scholes result: under \(\mathbb{Q}\), \(\log S_T \sim N(x + (r - \frac{1}{2}\sigma^2)\tau, \sigma^2\tau)\), and the discount factor contributes \(e^{-r\tau}\).

General Payoffs via Fourier Inversion¶

The Feynman-Kac framework handles not only exponential payoffs \(h(x) = e^{\langle u, x \rangle}\) but also general payoffs through the Fourier representation:

\[ V(t, x) = \mathbb{E}\!\left[e^{-\int_t^T r_s\,ds}\,h(X_T) \mid X_t = x\right] \]

If \(h\) has a Fourier transform \(\hat{h}(v) = \int e^{ivx}h(x)\,dx\), then:

\[ V(t, x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{h}(v)\,\mathcal{T}(\tau, -iv, x)\,dv \]

where \(\mathcal{T}\) is the discounted transform. Since \(\mathcal{T}\) has exponential-affine form, the only numerical step is evaluating the Fourier integral.

PDE vs ODE Dimensionality

The Feynman-Kac PDE is \((d+1)\)-dimensional (time plus \(d\) spatial dimensions). Finite difference methods have cost \(O(N^d)\) where \(N\) is the grid size per dimension. The affine Riccati approach replaces this with a system of \(d+1\) ODEs, each one-dimensional. For \(d = 5\) (a five-factor model), the PDE approach is computationally prohibitive, while the ODE approach remains trivial. This dimensional reduction is the fundamental reason affine models are preferred for multi-factor applications.

Summary¶

The Feynman-Kac theorem connects the discounted conditional expectation \(V(t, x) = \mathbb{E}[e^{-\int_t^T c(X_s)\,ds}\,h(X_T) \mid X_t = x]\) to the PDE \(\partial_t V + \mathcal{A}V - cV = 0\). For affine processes with exponential terminal conditions, the exponential-affine ansatz reduces this PDE to the extended Riccati system \(\tilde{\phi}' = F(\tilde{\psi}) - \rho_0\), \(\tilde{\psi}' = R(\tilde{\psi}) - \rho_1\). This reduction from PDE to ODE is the core computational advantage of the affine framework, and it extends to general payoffs through Fourier inversion of the discounted characteristic function.

Exercises¶

Exercise 1. State the Feynman-Kac theorem for the function \(V(t, x) = \mathbb{E}^{\mathbb{Q}}[e^{-\int_t^T r(X_s)\,ds}\,h(X_T) \mid X_t = x]\). Write down the PDE that \(V\) satisfies, including the terminal condition. For the Black-Scholes model with \(h(x) = (e^x - K)^+\), identify the generator \(\mathcal{A}\) and the discount rate \(r\).

Solution to Exercise 1

Feynman-Kac Theorem. Let \(X_t\) be an Ito diffusion with generator \(\mathcal{A}\) and let \(r(x)\) be the short rate. Define

\[ V(t, x) = \mathbb{E}^{\mathbb{Q}}\!\left[e^{-\int_t^T r(X_s)\,ds}\,h(X_T) \;\middle|\; X_t = x\right] \]

Under suitable regularity conditions, \(V\) is the unique classical solution of the PDE

\[ \frac{\partial V}{\partial t}(t, x) + \mathcal{A}V(t, x) - r(x)\,V(t, x) = 0, \qquad V(T, x) = h(x) \]

For the Black-Scholes model with \(X_t = \log S_t\), the dynamics are \(dX_t = (r - \frac{1}{2}\sigma^2)\,dt + \sigma\,dW_t\). The generator is

\[ \mathcal{A} = (r - \tfrac{1}{2}\sigma^2)\frac{\partial}{\partial x} + \tfrac{1}{2}\sigma^2\frac{\partial^2}{\partial x^2} \]

The discount rate is the constant \(r(x) = r\), and the terminal condition for a call is \(h(x) = (e^x - K)^+\). The Feynman-Kac PDE therefore reads

\[ \frac{\partial V}{\partial t} + (r - \tfrac{1}{2}\sigma^2)\frac{\partial V}{\partial x} + \tfrac{1}{2}\sigma^2\frac{\partial^2 V}{\partial x^2} - rV = 0, \qquad V(T, x) = (e^x - K)^+ \]

Exercise 2. For a one-dimensional affine diffusion \(dX_t = (\kappa_0 + \kappa_1 X_t)\,dt + \sqrt{\sigma_0 + \sigma_1 X_t}\,dW_t\) with short rate \(r(x) = \rho_0 + \rho_1 x\), substitute the exponential-affine ansatz \(V(\tau, x) = e^{\phi(\tau) + \psi(\tau)x}\) into the Feynman-Kac PDE \(\frac{\partial V}{\partial \tau} = \mathcal{A}V - rV\) and derive the extended Riccati system by matching constant and linear terms in \(x\).

Solution to Exercise 2

The generator for the one-dimensional affine diffusion is

\[ \mathcal{A} = (\kappa_0 + \kappa_1 x)\frac{\partial}{\partial x} + \frac{1}{2}(\sigma_0 + \sigma_1 x)\frac{\partial^2}{\partial x^2} \]

With \(r(x) = \rho_0 + \rho_1 x\), the Feynman-Kac PDE in backward time \(\tau = T - t\) is

\[ \frac{\partial V}{\partial \tau} = (\kappa_0 + \kappa_1 x)\frac{\partial V}{\partial x} + \frac{1}{2}(\sigma_0 + \sigma_1 x)\frac{\partial^2 V}{\partial x^2} - (\rho_0 + \rho_1 x)V \]

Substituting the ansatz \(V = e^{\phi(\tau) + \psi(\tau)x}\), the derivatives are \(\frac{\partial V}{\partial \tau} = (\phi' + \psi' x)V\), \(\frac{\partial V}{\partial x} = \psi V\), and \(\frac{\partial^2 V}{\partial x^2} = \psi^2 V\). Dividing through by \(V > 0\):

\[ \phi' + \psi' x = (\kappa_0 + \kappa_1 x)\psi + \frac{1}{2}(\sigma_0 + \sigma_1 x)\psi^2 - \rho_0 - \rho_1 x \]

Constant terms (\(x^0\)):

\[ \phi'(\tau) = \kappa_0 \psi + \frac{1}{2}\sigma_0 \psi^2 - \rho_0 \]

Linear terms (coefficient of \(x\)):

\[ \psi'(\tau) = \kappa_1 \psi + \frac{1}{2}\sigma_1 \psi^2 - \rho_1 \]

This is the extended Riccati system. The \(\psi\)-equation is a scalar Riccati ODE with constant term \(-\rho_1\), linear term \(\kappa_1\), and quadratic coefficient \(\frac{1}{2}\sigma_1\). Once \(\psi(\tau)\) is obtained, \(\phi(\tau)\) is found by direct integration.

Exercise 3. Verify the Feynman-Kac solution for the Vasicek bond price. Starting from \(V(\tau, x) = P(t, T) = e^{A(\tau) + B(\tau)x}\) with the Vasicek generator, show that \(V\) satisfies \(\frac{\partial V}{\partial \tau} = \kappa(\theta - x)\frac{\partial V}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2 V}{\partial x^2} - xV\) by computing each derivative and substituting.

Solution to Exercise 3

For the Vasicek model \(dr_t = \kappa(\theta - r_t)\,dt + \sigma\,dW_t\), the generator is \(\mathcal{A} = \kappa(\theta - x)\frac{\partial}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2}{\partial x^2}\) and the killing rate is \(c(x) = x\).

With \(V = e^{A(\tau) + B(\tau)x}\), compute each term:

\(\frac{\partial V}{\partial \tau} = (A' + B' x)V\)
\(\frac{\partial V}{\partial x} = BV\)
\(\frac{\partial^2 V}{\partial x^2} = B^2 V\)

Substituting into \(\frac{\partial V}{\partial \tau} = \mathcal{A}V - xV\):

\[ (A' + B'x)V = \kappa(\theta - x)BV + \frac{1}{2}\sigma^2 B^2 V - xV \]

Dividing by \(V > 0\) and collecting terms:

Constant terms: \(A' = \kappa\theta B + \frac{1}{2}\sigma^2 B^2\)

Coefficient of \(x\): \(B' = -\kappa B - 1\)

The \(B\)-equation \(B' = -\kappa B - 1\) with \(B(0) = 0\) has solution \(B(\tau) = -\frac{1 - e^{-\kappa\tau}}{\kappa}\). One can verify: \(B'(\tau) = -e^{-\kappa\tau}\) and \(-\kappa B(\tau) - 1 = -\kappa \cdot \left(-\frac{1 - e^{-\kappa\tau}}{\kappa}\right) - 1 = (1 - e^{-\kappa\tau}) - 1 = -e^{-\kappa\tau}\), confirming equality. Also \(B(0) = -\frac{1 - 1}{\kappa} = 0\), matching the initial condition.

Exercise 4. Explain why the Feynman-Kac approach converts the computational cost from "exponential in dimension \(d\)" (for PDE grid methods) to "polynomial in \(d\)" (for Riccati ODE methods) when the process is affine. What is the dominant cost in the Riccati approach as the number of factors \(d\) increases?

Solution to Exercise 4

PDE grid methods discretize the spatial domain on a grid with \(N\) points per dimension. For a \(d\)-dimensional PDE, the total number of grid points is \(N^d\). Each time step of an implicit finite difference scheme requires solving a linear system of size \(N^d\), giving cost \(O(N^d)\) per time step and \(O(N_\tau \cdot N^d)\) total. For \(d = 5\) and \(N = 100\), this is \(10^{10}\)---prohibitively expensive.

Riccati ODE methods solve a system of \(d + 1\) scalar ODEs (one for \(\tilde{\phi}\) and \(d\) for \(\tilde{\psi}_1, \ldots, \tilde{\psi}_d\)). Each ODE is one-dimensional, and standard ODE solvers (e.g., Runge-Kutta) have cost \(O(N_\tau)\) per equation, giving total cost \(O((d+1) \cdot N_\tau)\)---linear in \(d\).

The dominant cost as \(d\) increases is the evaluation of the quadratic form \(\frac{1}{2}\langle \tilde{\psi}, a_j \tilde{\psi} \rangle\) in each \(\tilde{\psi}_j\)-equation, which involves \(O(d^2)\) operations per evaluation (matrix-vector product). Thus the total cost scales as \(O(d^2 \cdot N_\tau)\), which is polynomial in \(d\) rather than exponential.

Exercise 5. The Feynman-Kac theorem requires regularity conditions on the coefficients and the terminal function \(h\). For the payoff \(h(x) = (e^x - K)^+\), explain why \(h\) is not smooth at \(x = \log K\) and discuss how this affects the validity of the Feynman-Kac representation. Does the exponential-affine form still apply?

Solution to Exercise 5

The payoff \(h(x) = (e^x - K)^+\) has a kink at \(x = \log K\): it equals \(0\) for \(x < \log K\) and \(e^x - K\) for \(x \geq \log K\). The function is continuous but not differentiable at \(x = \log K\) (the left derivative is \(0\) and the right derivative is \(K\)), so \(h\) is not a \(C^2\) function.

The classical Feynman-Kac theorem requires \(h\) to have sufficient smoothness (or polynomial growth conditions) for the solution \(V(t, x)\) to be a classical solution of the PDE. Since \(h\) is not smooth at \(x = \log K\), the PDE solution \(V(t, x)\) may not be classical at \(t = T\). However, for \(t < T\) (strictly before maturity), the diffusion process smooths the terminal condition, and \(V(t, x)\) is indeed \(C^{1,2}\) and satisfies the PDE classically.

The exponential-affine form \(V = e^{\tilde{\phi} + \langle \tilde{\psi}, x \rangle}\) does not directly apply to the call payoff because this ansatz corresponds to the specific terminal condition \(h(x) = e^{\langle u, x \rangle}\), not \((e^x - K)^+\). To price the call, one uses Fourier inversion: express \(h\) as an integral of exponential functions, apply the exponential-affine formula to each, and integrate. The Feynman-Kac representation remains valid (in the mild/viscosity sense), but the affine form is used indirectly through the characteristic function, not directly for the call payoff.

Exercise 6. For a two-factor affine model with state vector \((r_t, V_t)\) and short rate \(r(x) = x_1\), write down the Feynman-Kac PDE in two spatial dimensions. Show that the exponential-affine ansatz \(V(\tau, x_1, x_2) = e^{\phi(\tau) + \psi_1(\tau)x_1 + \psi_2(\tau)x_2}\) reduces this PDE to a system of three ODEs.

Solution to Exercise 6

Let the two-factor affine model have state \((X_t^{(1)}, X_t^{(2)}) = (r_t, V_t)\) with dynamics

\[ dX^{(1)} = b_1(x)\,dt + \sigma_{11}(x)\,dW_1 + \sigma_{12}(x)\,dW_2 \]

\[ dX^{(2)} = b_2(x)\,dt + \sigma_{21}(x)\,dW_1 + \sigma_{22}(x)\,dW_2 \]

With \(r(x) = x_1\) (so \(\rho_0 = 0\), \(\rho_1 = (1, 0)^T\)), the Feynman-Kac PDE in two spatial dimensions is

\[ \frac{\partial V}{\partial \tau} = b_1(x)\frac{\partial V}{\partial x_1} + b_2(x)\frac{\partial V}{\partial x_2} + \frac{1}{2}a_{11}(x)\frac{\partial^2 V}{\partial x_1^2} + a_{12}(x)\frac{\partial^2 V}{\partial x_1 \partial x_2} + \frac{1}{2}a_{22}(x)\frac{\partial^2 V}{\partial x_2^2} - x_1 V \]

where \(a_{ij}(x) = \sum_k \sigma_{ik}(x)\sigma_{jk}(x)\) are the diffusion matrix entries, all affine in \(x\).

Substituting the ansatz \(V = e^{\phi(\tau) + \psi_1(\tau)x_1 + \psi_2(\tau)x_2}\) and dividing by \(V\):

\[ \phi' + \psi_1' x_1 + \psi_2' x_2 = b_1 \psi_1 + b_2 \psi_2 + \frac{1}{2}a_{11}\psi_1^2 + a_{12}\psi_1\psi_2 + \frac{1}{2}a_{22}\psi_2^2 - x_1 \]

Since all coefficients are affine in \((x_1, x_2)\), collecting constant, \(x_1\)-, and \(x_2\)-terms yields three ODEs:

ODE for \(\phi\): \(\phi'(\tau) = F(\psi_1, \psi_2) - 0\) (constant terms)

ODE for \(\psi_1\): \(\psi_1'(\tau) = R_1(\psi_1, \psi_2) - 1\) (coefficient of \(x_1\), with \(-\rho_{1,1} = -1\))

ODE for \(\psi_2\): \(\psi_2'(\tau) = R_2(\psi_1, \psi_2) - 0\) (coefficient of \(x_2\), with \(-\rho_{1,2} = 0\))

This is a system of three scalar ODEs (two coupled for \(\psi_1, \psi_2\), plus one quadrature for \(\phi\)), replacing the original three-dimensional PDE (\(\tau, x_1, x_2\)). The initial conditions are \(\phi(0) = 0\), \(\psi_1(0) = u_1\), \(\psi_2(0) = u_2\).