A BSDE approach to fair bilateral pricing under endogenous collateralization

Nie and Rutkowski [21, 22] examined fair bilateral pricing in models with funding costs and an exogenously given collateral. The goal of this work is to extend their results to the case of an endogenous margin account, that is, the collateral that may depend on the contract’s value for the hedger and/or the counterparty. Comparison theorems for BSDEs from Nie and Rutkowski [23] are used to derive the bounds for unilateral prices and to study the range for fair bilateral prices in a general semimartingale model. For the case of the negotiated collateral, the backward stochastic viability property from Buckdahn et al. [7] is employed to examine the bounds for fair bilateral prices of European claims in a diffusion-type model. As a by-product, we generalize in several respects the option pricing results from Bergman [1], Mercurio [19] and Piterbarg [27]. First, we consider general collateralized contracts with a stream of cash flows, rather than path-independent European claims. Second, we examine not only the case where the collateral is set by one party, but also the case of a collateral negotiated between the counterparties. Third, we study not only the Bergman model with differing lending and borrowing cash rates, but also a trading model with idiosyncratic funding costs for risky assets.