The recent financial crisis has highlighted the importance to account for counterparty risk and funding costs in the valuation of over-the-counter portfolios of derivatives. When managing their portfolios, traders face costs for maintaining the hedge of the position, posting collateral resources, and servicing their collateral requests. Due to the interdependencies between these operations, such costs cannot be separated and attributed to different business units (CVA, DVA, and EVA desks). In this talk, we introduce a unified framework for computing the total costs, referred to as XVA, of an European style derivative transaction traded between two risky counterparties. We use no-arbitrage arguments to derive the nonlinear backward stochastic differential equations (BSDEs) associated with the portfolios which replicate long and short positions in the claim. This leads to defining buyer's and seller's XVAs which in turn identify a no-arbitrage band. When borrowing and lending rates coincide, our framework recovers a generalized version of Piterbarg's model. In this case, we provide a fully explicit expression for the uniquely determined price of XVA. When they differ, we derive the semi-linear partial differential equations (PDEs) associated with the non-linear BSDEs and show that they admit a unique classical solution. We use these solutions to conduct a numerical analysis showing high sensitivity of the no-arbitrage band and replicating strategies to funding spreads and collateral levels. Based on joint work with Stephan Sturm and Maxim Bichuch.
Biography
Agostino Capponi is an assistant professor in the IEOR Department at Columbia University, where he is also a member of the Institute for Data Science and Engineering. Agostino received his Master and Ph.D. Degree in Computer Science and Applied and Computational Mathematics from the California Institute of Technology, respectively in 2006 and 2009. His main research interests are in the area of networks, with a special focus on systemic risk, contagion, and control. In the context of financial networks, the outcome of his research contributes to a better understanding of risk management practices, and to assess the impact of regulatory policies aimed at controlling financial markets. He has been awarded a grant from the Institute for New Economic Thinking for his research on dynamic contagion mechanisms. His work on systemic risk dynamics under central clearing done in collaboration with the Department of Treasury has obtained press coverage from major organizations such as Bloomberg and Reuters. His research has been published in top-tier journals of Financial Mathematics, Operations Research, and Engineering. His work as also been published in leading practitioner journals and invited book chapters. Agostino holds as world patent for a target tracking methodology in military networks.
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