Abstracts

We have recently observed the traditional derivative pricing framework being extended to include several adjustments. These adjustments involving for instance the recognition of counterparty credit risk or funding costs were intended to fix the assumptions of the prevailing models that failed to hold in the last global financial crisis.
In the last two years, the Funding Value Adjustment (FVA) acquired greater significance, being the subject of many papers and academic discussions. However, the insertion of this adjustment in particular and of the funding costs more generally, adds another layer of complexity to the valuation of contingent claims given the recursive nature of funding decisions and therefore requires a delicate treatment using robust definitions.
In the present work, we construct a formal mathematical framework to be utilized in the valuation of derivative instruments in a market under funding costs, consequently enabling an explicit formulation for FVA. To achieve this goal, we make use of the well known martingale approach customizing it to fit our purpose.
We show that even under different funding accounts the arguments behind the market price of risk still hold in nature but at the same time old formulations are not longer valid. Although the martingale setting is still useful for the problem at hand many aspects of the classical risk-neutral valuation framework will suffer modifications. These changes affect for example the properties of the numeraires, the specification of the state price densities and of course the representation of the equivalent martingale measures.
Making use of our framework we prove that many formulations for the FVA present in the academic literature, while comparable at first glance, are truthfully inconsistent when strictly analysed. In order to examine our setting in a practical context, we apply our theoretical approach to interest rate derivatives obtaining expressions for the FVA of those instruments.