Valuation and Hedging of Contracts with Funding Costs and Collateralization

TL;DR

This paper develops a unified theoretical framework for the valuation and hedging of OTC financial contracts in nonlinear markets, accounting for funding costs, collateralization, and credit risk, extending existing models with a systematic approach.

Contribution

It introduces a comprehensive approach to nonlinear market valuation, defining no-arbitrage conditions and deriving non-linear BSDEs for pricing in complex financial environments.

Findings

Identifies primary sources of valuation adjustments.

Provides conditions for absence of arbitrage in nonlinear markets.

Derives non-linear BSDEs for no-arbitrage pricing.

Abstract

The research presented in this work is motivated by recent papers by Brigo et al. (2011), Burgard and Kjaer (2009), Cr\'epey (2012), Fujii and Takahashi (2010), Piterbarg (2010) and Pallavicini et al. (2012). Our goal is to provide a sound theoretical underpinning for some results presented in these papers by developing a unified framework for the non-linear approach to hedging and pricing of OTC financial contracts. We introduce a systematic approach to valuation and hedging in nonlinear markets, that is, in markets where cash flows of the financial contracts may depend on the hedging strategies. Our systematic approach allows to identify primary sources of and quantify various adjustment to valuation and hedging, primarily the funding and liquidity adjustment and credit risk adjustment. We propose a way to define no-arbitrage in such nonlinear markets, and we provide conditions that imply absence of arbitrage in some specific market trading models. Accordingly, we formulate a concept of no-arbitrage price, and we provide relevant (non-linear) BSDE that produces the no-arbitrage price in case when the contract's cash flows can be replicated.