On Funding Costs and the Valuation of Derivatives

Abstract

Since the credit crisis the valuation of simple derivatives has become much more complex, primarily through so-called adjustments such as the credit value, debt value and funding value adjustment. Most of these elements of the valuation are well understood, although not always easily calculated. However, on the inclusion of the funding costs in the valuation, through a funding value adjustment, there is currently no consensus. This paper shows that the lack of concensus comes from two fundamentally different assumptions regarding funding costs of a bank. One assumption, mostly used in the literature, is that funding costs are fixed. This leads to derivatives' values that, in general, depend on the funding rate of the bank. The other, newly introduced, assumption is that funding costs immediately adjust after each new transaction to reflect the new asset composition. It is shown that under this assumption, in the Black-Scholes model, the funding costs of the bank do not affect the value of derivatives. I argue that the latter assumption is more appropriate for the valuation of derivatives.