No abstract

The research presented in this work is motivated by recent papers by Brigo et al. [5,6], Burgard and Kjaer [7,8,10], Crépey [14,15], Fujii and Takahashi [21], Piterbarg [38] and Pallavicini et al. [37]. 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. The impact that various funding bases and margin covenants exert on the values and hedging strategies for OTC contracts is examined. The relationships between our research and papers by other authors, with an exception of Pallavicini et al. [37] and Piterbarg [38], are not discussed in this part of our research. More detailed studies of these relationships, as well as the issue of the counterparty credit risk, are examined in the follow-up paper.

Bielecki and Rutkowski introduced and studied a generic nonlinear market model, which includes several risky assets, multiple funding accounts, and margin accounts. In this paper, we examine the pricing and hedging of contract from the perspective of both the hedger and the counterparty with arbitrary initial endowments. We derive inequalities for unilateral prices and we study the range of fair bilateral prices. We also examine the positive homogeneity and monotonicity of unilateral prices with respect to the initial endowments. Our study hinges on results from Nie and Rutkowski for backward stochastic differential equations (BSDEs) driven by continuous martingales, but we also derive the pricing partial differential equations (PDEs) for path‐independent contingent claims of a European style in a Markovian framework.

We establish some well-posedness and comparison results for BSDEs driven by one-and multi-dimensional martingales. On the one hand, our approach is largely motivated by results and methods developed in Carbone et al. [3] and El Karoui and Huang [7]. On the other hand, our results are also motivated by the recent developments in arbitrage pricing theory under funding costs and collateralization. A new version of the comparison theorem for BSDEs driven by a multi-dimensional martingale is established and applied to the pricing and hedging BSDEs studied in Bielecki and Rutkowski [1] and Nie and Rutkowski [25]. This allows us to obtain the existence and uniqueness results for unilateral prices and to demonstrate the existence of no-arbitrage bounds for a collateralized contract when both agents have non-negative initial endowments.