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

TL;DR

This paper introduces a fixed point method for valuing derivatives with counterparty risk and collateralization, providing a reliable iterative scheme and analyzing how risk factors influence bid-ask spreads.

Contribution

It develops a novel fixed point approach for derivative valuation under counterparty risk, ensuring unique solutions and enabling efficient numerical computation.

Findings

The fixed point approach guarantees a unique, bounded, continuous solution.

The iterative scheme converges to the true derivative price.

Counterparty risk and collateralization significantly affect bid-ask spreads.

Abstract

This paper studies a valuation framework for financial contracts subject to reference and counterparty default risks with collateralization requirement. We propose a fixed point approach to analyze the mark-to-market contract value with counterparty risk provision, and show that it is a unique bounded and continuous fixed point via contraction mapping. This leads us to develop an accurate iterative numerical scheme for valuation. Specifically, we solve a sequence of linear inhomogeneous PDEs, whose solutions converge to the fixed point price function. We apply our methodology to compute the bid and ask prices for both defaultable equity and fixed-income derivatives, and illustrate the non-trivial effects of counterparty risk, collateralization ratio and liquidation convention on the bid-ask spreads.