Feynman–Kac Representation of Counterparty and Funding Risk PDEs

Summary

The document asks why the Feynman–Kac theorem can be applied to a derivatives valuation partial differential equation that includes minimum and maximum terms. The displayed equation models an adjustment value with bilateral counterparty default intensities, recovery rates, and a funding spread, subject to a zero terminal condition. Its spatial differential operator describes the underlying asset dynamics through volatility and a drift adjusted for payouts or financing effects.

The question contrasts this nonlinear-looking equation with a familiar Feynman–Kac form containing a source term and a linear discount term. It seeks an explanation of how the maximum and minimum expressions can be handled, or a reference to a version of the theorem that fits the equation. The document provides no answer, derivation, or citation beyond identifying the equation and its context. It is useful as a focused mathematical question about applying stochastic representation to counterparty-risk valuation, but it does not establish the required conditions or explain the transformation; those depend on the precise treatment of the nonlinear terms and solution assumptions.

Key ideas

• The valuation PDE includes counterparty default and funding adjustments expressed through positive and negative exposure terms.
• The underlying operator captures asset price diffusion and adjusted drift.
• The question is how Feynman–Kac applies when the PDE contains maximum and minimum functions.
• No derivation or resolution is provided, so applicability conditions remain open.

Tags

# Why is Feynman-Kac formula applicable in Burgard-Kjaers PDE paper?


# Why is Feynman-Kac formula applicable in Burgard-Kjaers PDE paper?












In the paper Partial Differential Equation Representation of Derivatives with Bilateral Counterparty Risk and Funding Costs by Burgard and Kjaer, they say we may formally apply the Feynman-Kac theorem to equation 29 which gives equation 30.

Equation 29 in the article is:

$\delta_{t} U + A_tU - rU = (1-R_{B})\lambda_B \text{min}(V+U,0) + (1-R_{C})\lambda_C \text{max}(V+U,0) + s_F \text{max}(V+U,0),$

with boundary condition $U(T,S) = 0$.

Here $A_t$ is the differential operator given by:

$A_t V = \frac{1}{2} \sigma^2 S^2 \partial^2_{S} V + (q_s - \gamma_s) S \partial_S V$.

Not caring about technical conditions concerning function growth and similar in Feynman-Kac formula, what is the motivation that the Feynman-Kac formula is applicable here?

The Feynman-Kac formula is usually stated for the case of a function $U$, where we don't have any maximum or minimum functions applied.

E.g. in Karatzas-Shreve: Brownian Motion and Stochastic Calculus on page 268 we have this version:

$\partial_{t} U + \frac{1}{2}\partial_{S}^2 U - kU= g$.

How it is possible to transform the equation 29 such that it can be applied to the Feynman-Kac formula as stated e.g. in Karatzas-Shreve's book? Or where can you find a reference to a version of Feynman-Kac formula which closely resembles equation 29?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.