Self-Financing Replication and Funding Adjustments in Derivative Valuation
Summary
The document explains the link between self-financing replication and funding constraints in derivative valuation under counterparty credit and funding risk. A self-financing portfolio cannot receive or return external cash while it is being managed, so its hedge positions must account for the cash needed to support the derivative and its collateral. When collateralization is incomplete, the replicated economic value can differ from the idealized risk-free value.
The response describes that gap as reflecting residual counterparty, own-default, and funding risks. In an incomplete market, available instruments may not permit those exposures to be fully hedged; the resulting costs appear in valuation adjustments such as CVA, DVA, and FVA. The discussion is conceptual and gives no equations, worked replication portfolio, or quantitative evidence. It also does not detail how each adjustment is calculated or how a particular funding constraint determines hedge positions.
Key ideas
- A self-financing replication portfolio cannot rely on external cash injections or withdrawals during its construction.
- Incomplete collateralization can make the economic derivative value differ from its risk-free value.
- Credit and funding exposures may remain unhedged when the market lacks sufficient instruments.
- Residual costs may be represented through CVA, DVA, and FVA valuation adjustments.
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Full text
# How is the self-financing constraint linked to funding instruments in Burgard & Kjaer
# How is the self-financing constraint linked to funding instruments in Burgard & Kjaer
In Burgard & Kjaer (2011), the replication of a derivative under credit and funding risk requires satisfying a self-financing condition, and also accounting for the difference between the risk-free value $V$ and the actual value $\hat{V}$ due to incomplete collateralization or credit/funding effects.
> "With a single bond, once the funding constraint is fulfilled, there are no degrees of freedom left for the issuer to hedge out his own default."
I understand that when the position is not fully collateralized, and we exit a world without credit risk, there is a need to fund the difference between the economic value and collateral. This difference cannot be absorbed without trading instruments that shift value dynamically.
How is the self-financing condition directly related to the funding constraint in the replication portfolio?
## Answer by Marco (score 1)
https://quant.stackexchange.com/a/83847
The difference between $\hat{V}$ and $V$ reflects the cost of replicating the derivative in the presence of incomplete collateralization and funding constraints. It can be interpreted as the value of residual risks (e.g., counterparty risk, own default risk, and funding risk) that would require additional instruments (e.g., CCDS) to hedge in a complete market.
In a self-financing setup, where no external capital can be injected or withdrawn, these risks cannot always be fully hedged using the available instruments. As a result, the associated costs are captured through valuation adjustments such as CVA, DVA, and FVA.
Because the market is incomplete and collateralization is not perfect, full replication is not possible. Therefore, the economic value $\hat{V}$ departs from the idealized risk-free value $V$, and this difference quantifies the cost of bearing those unhedged risks.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.