Funding Cost Adjustment and Default Loss in Derivative Hedging
Summary
The document examines how a funding cost adjustment (FCA) arises in Burgard and Kjaer’s framework for pricing derivatives with counterparty and issuer default risk. It quotes an FCA expression whose integrand depends on the issuer’s default intensity, recovery, discounting, and expected positive derivative value. The accompanying explanation describes a default scenario in which unsecured funding supports a positive close-out amount and the funder recovers only part of its claim, creating a loss.
The author asks how this account relates to a passage describing FCA for an out-of-the-money position that requires funding. They propose that the funding strategy and issuer bonds used in hedging might explain the apparent mismatch, but provide no resolution or supporting analysis. The note is therefore a conceptual question about sign conventions, funding balances, and hedging assumptions, rather than a settled derivation; its conclusions should not be treated as established guidance.
Key ideas
- The quoted FCA expression links funding adjustment to issuer default risk, recovery, discounting, and expected positive derivative value.
- Unsecured funding can create a loss when the issuer defaults and the funder recovers less than the amount owed.
- The document questions how FCA can also apply to an out-of-the-money position that requires funding.
- The proposed role of issuer bonds in hedging is a hypothesis, not a conclusion established by the document.
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# FCA (Funding Cost Adjustment) in Burgard & Kjaer
# FCA (Funding Cost Adjustment) in Burgard & Kjaer
From "In the Balance" of Burgard, Kjaer
> $$ \mathrm{FCA} \;=\; -(1-R_B)\int_t^T \lambda_B(u)\,D_{r+\lambda_B+\lambda_C}(t,u)\, \mathbb{E}_t\big[\,V^+(u,S(u))\,\big]\,\mathrm{d}u. $$ To understand the origin of this term, it is instructive to see what happens when the issuer defaults while $\hat{V}$ is positive (i.e. in-the-money for the issuer). In this case, just prior to default the cash account $\beta_F$ is negative and its corresponding amount $-V^+$ is provided unsecured by the funding provider. If the issuer then defaults, the derivatives desk will settle the derivative at the close-out amount and receive $V^+$. The external, unsecured funding provider, on the other hand, will only recover $R_B\,V^+$ on its funding, thereby suffering a loss of $(1-R_B)V^+$, which is the windfall priced by the FCA integrand above.
$$ \underbrace{\alpha_B= - \frac{U+(1-R_B)V^-}{P_B}}_{\text{units of zero-recovery zero-coupon-bond of the issuer}} $$ $$ \beta_F = -\hat{V}-\alpha_BP_B = -V^+ -R_B V^- $$
i.e. it seems to me that FCA arises while $\hat{V}$ is positive (i.e. in-the-money for the issuer) and funding is required due to negative cash account $\beta_F$. What puzzles me is that at the beginning of "Funding Costs, Funding Strategies"
> When it is out-of-the money and requires funding, a post-default windfall to the issuer's estate is generated. In that case, a funding cost adjustment (FCA) is added in to compensate.
How does this final phrase link to the initial explanation? Under what scenario would an out-of-the-money position for the issuer require unsecured funding, thereby generating a Funding Cost Adjustment (FCA)? The authors state that an FCA arises for an in-the-money position because it creates an unsecured liability ($\beta_F = -V^+$). However, they also claim an FCA can arise from an out-of-the-money position that "requires funding."
My hypothesis is that this is not about the derivative's moneyness itself, but about the specific funding strategy chosen for the hedging portfolio. The portfolio is funded using the issuer's own bonds ($P_B$), which inherently embed their credit risk. Therefore, even a position that is theoretically "funded" by a positive cash balance might still be exposed to the issuer's credit spread through the hedging instruments, effectively creating a funding cost.
Am I correct in interpreting that the FCA for an OTM position stems from the credit risk of the bonds used to hedge the portfolio, rather than from borrowing cash to cover a negative balance (collateral which is related to COLVA)? Or am I missing a more fundamental mechanism?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.