Pricing Derivatives with Counterparty Risk and Collateralization: A Fixed Point Approach
European Journal of Operations Research, 2015, Forthcoming
28 Pages Posted: 16 Nov 2013 Last revised: 23 Jun 2015
Date Written: January 25, 2015
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.
Keywords: bilateral counterparty risk, collateralization, credit valuation adjustment, fixed point method, contraction mapping
JEL Classification: G12, G13, G23, C63
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