Finance and Stochastics

, Volume 20, Issue 4, pp 855–900 | Cite as

A BSDE approach to fair bilateral pricing under endogenous collateralization



Nie and Rutkowski (Int. J. Theor. Appl. Finance 18:1550048, 2015; Math. Finance, 2016, to appear) examined fair bilateral pricing in models with funding costs and an exogenously given collateral. The main goal of this work is to extend results from Nie and Rutkowski (Int. J. Theor. Appl. Finance 18:1550048, 2015; Math. Finance, 2016, to appear) to the case of an endogenous margin account depending on the contract’s value for the hedger and/or the counterparty. Comparison theorems for BSDEs from Nie and Rutkowski (Theory Probab. Appl., 2016, forthcoming) are used to derive bounds for unilateral prices and to study the range for fair bilateral prices in a general semimartingale model. The backward stochastic viability property, introduced by Buckdahn et al. (Probab. Theory Relat. Fields 116:485–504, 2000), is employed to examine the bounds for fair bilateral prices for European claims with a negotiated collateral in a diffusion-type model. We also generalize in several respects the option pricing results from Bergman (Rev. Financ. Stud. 8:475–500, 1995), Mercurio (Actuarial Sciences and Quantitative Finance, pp. 65–95, 2015) and Piterbarg (Risk 23(2):97–102, 2010) by considering contracts with cash-flow streams and allowing for idiosyncratic funding costs for risky assets.


Collateral Fair pricing Funding costs 

Mathematics Subject Classification (2010)

60G35 91G20 91G80 

JEL Classification




The research of T. Nie was supported by the Fundamental Research Fund of Shandong University (2015HW023). The research of M. Rutkowski was supported by the DVC Research Bridging Support Grant “A BSDE Approach to Models with Funding Costs”.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanChina
  2. 2.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia
  3. 3.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland

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