Pricing Derivatives with Counterparty Risk and Collateralization: A Fixed Point Approach

摘要

This paper studies a valuation framework for financial contracts subject toreference and counterparty default risks with collateralization requirement. Wepropose a fixed point approach to analyze the mark-to-market contract valuewith counterparty risk provision, and show that it is a unique bounded andcontinuous fixed point via contraction mapping. This leads us to develop anaccurate iterative numerical scheme for valuation. Specifically, we solve asequence of linear inhomogeneous PDEs, whose solutions converge to the fixedpoint price function. We apply our methodology to compute the bid and askprices for both defaultable equity and fixed-income derivatives, and illustratethe non-trivial effects of counterparty risk, collateralization ratio andliquidation convention on the bid-ask spreads.