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

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.

Summary

This paper identifies a specific mathematical error in the self-financing conditions stated in Piterbarg (2010) and Burgard & Kjaer (2011). The error arises from a common misapplication of the stochastic Leibniz rule: both papers fail to distinguish between price processes, dividend processes, and gain processes (in the sense of Duffie). The consequence is that a subportfolio (the equity position in Piterbarg, or the risky assets in Burgard-Kjaer) is implicitly assumed to be self-financing on its own, which would imply no rebalancing occurs — clearly wrong.

Crucially, the paper shows that the final pricing PDEs in both Piterbarg and Burgard-Kjaer are correct despite the erroneous self-financing condition. The error cancels in the derivation. The corrected self-financing condition uses gain processes $d{G}_i^A=d{P}_i^A+d{D}_i^A$ (price + dividends) and requires the full portfolio (not a subportfolio) to be self-financing.

Key Contributions

• Identifies the stochastic Leibniz rule error: $d(\Delta (t)S(t))\ne \Delta (t)dS(t)$ in general (boxed as WRONG)
• Shows the error appears in both Piterbarg (2010) and Burgard & Kjaer (2011) with the same structure
• Provides the corrected derivation using Duffie’s gain process framework
• Proves the final pricing PDE is unaffected — the error cancels

Key Findings

• The erroneous self-financing condition in Piterbarg: $d\gamma (t)=dV(t)-\Delta (t)dS(t)$ combined with $V=\Delta S+\gamma$ implies $d(\Delta S)=\Delta dS$ (equity is self-financing alone) — WRONG
• The corrected condition uses gain processes: $d{G}^{\Pi }=\sum {\theta }_{i}d{G}_i^A$ where the portfolio gain equals the weighted sum of individual asset gains
• The final Piterbarg PDE $\Delta (r_D-r_R)S+r_{C}C+r_F(V^Y-C)=V_t^Y+\frac{1}{2}{\sigma }_S^2{S}^2{V}_{SS}^Y$ is recovered from the corrected derivation

Critical Notes

Not the "+1 shift" error claimed in the XVA HJB paper

The error identified here is a stochastic Leibniz rule misapplication (confusing price and gain processes), NOT a “+1 shift” or collateral sign convention error. The XVA HJB paper’s Remark 5 claims “an incorrect $-1$ shift appears in the literature” regarding the collateral conjugate sign — this is a different issue from what Brigo et al. identify. Remark 5’s claim remains unverified against a specific source.

Important References

1. Piterbarg 2010 — “Funding beyond discounting” (contains the error, but correct final result)
2. Burgard and Kjaer 2011 — “PDE representations with bilateral counterparty risk and funding” (same error, same correct final result)
3. Duffie 2001 — Dynamic Asset Pricing Theory (gain process framework used for the correction)

Atomic Notes

• self-financing condition for funding and collateral

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