A BSDE approach to fair bilateral pricing under endogenous collateralization

TL;DR

This paper extends previous models of bilateral pricing to include endogenous collateralization, using BSDEs to derive fair price bounds and linking to market models with a single interest rate.

Contribution

It generalizes existing models by incorporating collateral that depends on contract value and applies BSDE comparison theorems to establish price inequalities.

Findings

Derived inequalities for unilateral prices.

Established bounds for fair bilateral prices.

Linked the model to single interest rate market.

Abstract

Our previous results are extended to the case of the margin account, which may depend on the contract's value for the hedger and/or the counterparty. The present work generalizes also the papers by Bergman (1995), Mercurio (2013) and Piterbarg (2010). Using the comparison theorems for BSDEs, we derive inequalities for the unilateral prices and we give the range for its fair bilateral prices. We also establish results yielding the link to the market model with a single interest rate. In the case where the collateral amount is negotiated between the counterparties, so that it depends on their respective unilateral values, the backward stochastic viability property studied by Buckdahn et al. (2000) is used to derive the bounds on fair bilateral prices.