A BSDE approach to fair bilateral pricing under endogenous collateralization

Abstract

Results from Nie and Rutkowski [12,14] are extended to the case of the margin account, which may depend on the contract's value for the hedger and/or the counterparty (recall that the collateral was given exogenously in [12,14]). The present work generalizes also the papers by Bergman [1], Mercurio [11] and Piterbarg [16]. Using the comparison theorems for BSDEs, we derive inequalities for the unilateral prices and we give the range for its fair bilateral prices. We also establish results yielding the link to the market model with a single interest rate. In the case where the collateral amount is negotiated between the counterparties, so that it depends on their respective unilateral values, the backward stochastic viability property studied by Buckdahn et al. [4] is used to derive the bounds on fair bilateral prices.

Arbitrage-free pricing of derivatives in nonlinear market models

“…Although Proposition 2 is fairly abstract, the sufficient condition formulated there can readily be verified, as soon as a specific market model is adopted (see, for instance, Bielecki and Rutkowski (2015) and Nie and Rutkowski (2015, 2016a (published online on 18 April 2017)). To support this claim, we will examine an example of a market model with idiosyncratic funding for risky assets and rehypothecated cash collateral.…”

“…Typically, a suitable variant of the Lipschitz continuity of a generator to a BSDE is sufficient to guarantee the desired properties of its solutions. Several instances of nonlinear market models with BSDEs satisfying the comparison property were studied by Nie and Rutkowski (2015, 2016a, although the concept of a regular model was not formally stated therein. In particular, they analyzed contracts with an endogenous collateral, meaning that an adjustment process X k explicitly depends on a solution Y (or even on solutions to the valuation problems for the hedger and the counterparty).…”

“…Let us stress that the disparity in unilateral valuation done by the two parties is a consequence of the nonlinearity of the wealth dynamics in trading strategies, so that it will typically occur even within the framework of a complete nonlinear model where the perfect replication of any contract can be achieved by the counterparties. The important issue of determining the range of fair bilateral prices in a general nonlinear framework is left for a future research (for results on bilateral pricing in some specific nonlinear models, we refer to Nie and Rutkowski (2015, 2016a.…”

“…Hence, in principle, it is possible to use our results to examine the interval of fair bilateral prices in a regular market model. 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).…”

Analysis of Nonlinear Valuation Equations Under Credit and Funding Effects

“…(10), another possible approach (without default risk) is shown for example in [9]. We introduce the process…”