Title: Pricing, Trading and Clearing of Defaultable Claims Subject to Counterparty Risk
Description: The recent financial crisis and subsequent regulatory changes on over-the-counter (OTC) markets have given rise to the new valuation and trading frameworks for defaultable claims to investors and dealer banks. More OTC market participants have adopted the new market conventions that incorporate counterparty risk into the valuation of OTC derivatives. In addition, the use of collateral has become common for most bilateral trades to reduce counterparty default risk. On the other hand, to increase transparency and market stability, the U.S and European regulators have required mandatory clearing of defaultable derivatives through central counterparties. This dissertation tackles these changes and analyze their impacts on the pricing, trading and clearing of defaultable claims. In the first part of the thesis, we study 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 partial differential equations, 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 prices. In the second part, we study the problem of pricing and trading of defaultable claims among investors with heterogeneous risk preferences and market views. Based on the utility-indifference pricing methodology, we construct the bid-ask spreads for risk-averse buyers and sellers, and show that the spreads widen as risk aversion or trading volume increases. Moreover, we analyze the buyer's optimal static trading position under various market settings, including (i) when the market pricing rule is linear, and (ii) when the counterparty -- single or multiple sellers -- may have different nonlinear pricing rules generated by risk aversion and belief heterogeneity.

For defaultable bonds and credit default swaps, we provide explicit formulas for the optimal trading positions, and examine the combined effect of heterogeneous risk aversions and beliefs. In particular, we find that belief heterogeneity, rather than the difference in risk aversion, is crucial to trigger a trade. Finally, we study the impact of central clearing on the credit default swap (CDS) market. Central clearing of CDS through a central counterparty (CCP) has been proposed as a tool for mitigating systemic risk and counterpart risk in the CDS market. The design of CCPs involves the implementation of margin requirements and a default fund, for which various designs have been proposed. We propose a mathematical model to quantify the impact of the design of the CCP on the incentive for clearing and analyze the market equilibrium. We determine the minimum number of clearing participants required so that they have an incentive to clear part of their exposures. Furthermore, we analyze the equilibrium CDS positions and their dependence on the initial margin, risk aversion, and counterparty risk in the inter-dealer market.

Our numerical results show that minimizing the initial margin maximizes the total clearing positions as well as the CCP's revenue.