Valuation and Hedging of Contracts with Funding Costs and Collateralization

Abstract

The research presented in this work is motivated by recent papers by Brigo, Capponi, and Pallavicini [Math. Finance, 24 (2014), pp. 125--46], Brigo et al. [Collateral Margining in Arbitrage-Free Counterparty Valuation Adjustment Including Re-hypothecation and Netting, preprint, arXiv:1101.3926, 2011], Burgard and Kjaer [Partial Differential Equation Representations of Options with Bilateral Counterparty Risk and Funding Costs, Working paper, 2009; J. Credit Risk, 7 (2011), pp. 1--19; Risk, December (2013), pp. 82--87], Crépey [Math. Finance, 25 (2015), pp. 1--22; Math. Finance, 25 (2015), pp. 23--50], Fujii and Takahashi [Quant. Finance, 13 (2013), pp. 749--768], Piterbarg [RISK, 24 (2012), pp. 97--102], and Pallavicini, Perini, and Brigo [Funding, Collateral and Hedging: Uncovering the Mechanics and the Subtleties of Funding Valuation Adjustments, preprint, arXiv:1210.3811, 2012]. Our goal is to provide a sound theoretical underpinning for some results presented in these papers by developing a unified framework for the nonlinear approach to hedging and pricing of OTC financial contracts. In particular, the impact that various funding bases and margin covenants exert on the values and hedging strategies for OTC contracts is examined.