- Fries, Christian P.: Discounting Revisited. Valuation under Funding, Counterparty Risk and Collateralization. 2010. (Download from SSRN)
- Fries, Christian P.: Funded Replication: Valuing with Stochastic Funding. 2011. (Download from SSRN)

Looking at the valuation of a swap when funding costs and counterparty risk are neglected (i.e., when there is a unique risk free discounting curve), it is natural to ask "What is the discounting curve of a swap in the presence of funding costs, counterparty risk and/or collateralization".

In this note we give an answer to this question. The answer is "There is no such thing as a unique discounting curve for swaps." In addition we derive some implied discounting when a deal is collateralized and make some concluding remarks on how credit valuation adjustments (CVAs) should be implemented.

The paper extends the considerations made in Section 28.4 "Receiversâ€™s and Payer's Credit Spreads" of the book Mathematical Finance: Theory, Modeling, Implementation.

Funding costs are the costs to a (risky) institution "A" of providing and managing its future cash flows in excess of, say, some risk free funding. For a single deterministic cash flow with maturity *T* these costs are essentially given by the ratio *P ^{A}(T) / P^{o}(T)* of the risky bond

For stochastic cash flows funding can be interpreted as a dynamic hedging of future cash flows. The problem can also lead to a complex portfolio problem, see [Discounting Revisited (2010)].

We will consider two possible valuations for a product consisting of future cash flows:

- The valuation which ignores any funding cost and calculates a liquidation value of the product. This valuation could be termed "mark-to-marked".
- The valuation including all funding costs from managing all future cash flows. This valuation could be termed "funded replication".

In this paper we will thoroughly introduce the valuation with funding, i.e. the "funded replication". We will also consider partial funding costs. For products with liquid market price we will derive a single consistent valuation: Having a liquid market price offers the option of canceling all future funding costs. This option can be valued.

The key points of this paper can be summarized as follows:

- The valuation of a derivative including funding corresponds to the valuation of a quanto product.
- Even a collateralized deal may bear some funding costs. These funding costs come from a correlation effect analog to the quanto adjustment.
- A liquid instrument having a market value represents the option to cancel all future funding. This option can be valued.