  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.
