Funding, repo and credit inclusive valuation as modified option pricing

Abstract

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 formula with dividends. This allows for a detailed valuation analysis, stress testing and risk analysis via sensitivities

Arbitrage-free pricing of derivatives in nonlinear market models

“…Particular instances of such unilateral and bilateral valuation problems were previously studied in Nie and Rutkowski (2015, 2016a (published online on 18 April 2017)) where it was shown that a non-empty interval of either fair bilateral prices or bilaterally profitable prices can be obtained in some nonlinear models for contracts with either an exogenous or an endogenous collateralization. It should be acknowledged that there exists a vast body of literature devoted to valuation and hedging of financial derivatives under differential funding costs, collateralization, the counterparty credit risk and other trading adjustments (see, for instance, Bichuch et al (2018), Brigo and Pallavicini (2014), Brigo et al (2018Brigo et al ( , 2017, Kjaer (2011, 2013), Crépey (2015a, b), Mercurio (2013), Pallavicini et al (2012b, a), and Piterbarg (2010). In view of limited space, we cannot present here these works in detail.…”

Binary funding impacts in derivative valuation

“…To avoid this difficulty, Brigo et al. (2017) considered uncollateralized contracts with null cash‐flow at defaults. On the other hand, Bichuch et al.…”

“…(2018) assumed a flexibility to choose a pricing measure for calculating closeout amount, and Brigo et al. (2017) considered only uncollateralized contracts with null cash‐flow at defaults. In the case of replacement cost , the mismatch does not appear, because, in our model, the funding rates are tacitly embedded in the replacement cost .…”

Risk-Neutral Valuation Under Differential Funding Costs, Defaults and Collateralization

“…We also demonstrate via an "invariance principle" that the risk-neutral valuation formula, which involves the risk-free rate often associated with a non-tradeable asset, can be cast in the replication approach in a valuation formula that eliminates all dependence on this illusory risk-free rate. For a practical example in this unified framework we refer to [9] where all the additional risk factors are integrated in the valuation without forcing separate adjustments to the price. For a specific contract this leads to the modified Black-Scholes pricing formula with dividends, which in turn allows for efficient sensitivity analysis.…”