Illustrating a problem in the self-financing condition in two 2010-2011 papers on funding, collateral and discounting

TL;DR

This paper identifies and clarifies a fundamental error in the self-financing condition used in two influential papers on funding and collateral in derivatives pricing, providing a corrected derivation.

Contribution

It highlights the specific mistake in the self-financing assumption and offers a clear derivation to correct the condition used in prior research.

Findings

The original papers' results remain valid despite the stated error.

The error stems from confusing price, dividend, and gain processes.

A corrected derivation clarifies the funding account structure.

Abstract

We illustrate a problem in the self-financing condition used in the papers "Funding beyond discounting: collateral agreements and derivatives pricing" (Risk Magazine, February 2010) and "Partial Differential Equation Representations of Derivatives with Counterparty Risk and Funding Costs" (The Journal of Credit Risk, 2011). These papers state an erroneous self-financing condition. In the first paper, this is equivalent to assuming that the equity position is self-financing on its own and without including the cash position. In the second paper, this is equivalent to assuming that a subportfolio is self-financing on its own, rather than the whole portfolio. The error in the first paper is avoided when clearly distinguishing between price processes, dividend processes and gain processes. We present an outline of the derivation that yields the correct statement of the self-financing condition, clarifying the structure of the relevant funding accounts, and show that the final result in "Funding beyond discounting" is correct, even if the self-financing condition stated is not.