Before revision: This paper specializes a number of earlier contributions to the theory of valuation of financial products in presence of credit risk, repurchase agreements and funding costs. Earlier works, including our own, pointed to the need of tools such as Backward Stochastic Differential Equations (BSDEs) or semi-linear Partial Differential Equations (PDEs), which in practice translate to ad-hoc numerical methods that are time-consuming and which render the full valuation and risk analysis difficult. We specialize here the valuation framework to benchmark derivatives and we show that, under a number of simplifying assumptions, the valuation paradigm can be recast as a Black-Scholes model with dividends.
After revision: We take the holistic approach of computing an OTC claim value that incorporates credit and funding liquidity risks and their interplays, instead of forcing individual price adjustments: CVA, DVA, FVA, KVA. The resulting nonlinear mathematical problem features semilinear PDEs and FBSDEs. We show that for the benchmark vulnerable claim there is an analytical solution, and we express it in terms of the Black-Scholes model with dividends.