Nonlinear Pricing in Discrete-time under Default and Optimal Collateral

Abstract

This thesis addresses a re-examination of the classical no-arbitrage pricing theory of mathematical finance through Backward Stochastic Difference Equations (BSdEs), their extensions and connections to Nonlinear Evaluations and Generalized Game Contingent Claims (GGCCs). A theory is developed in discrete-time encompassing the nonlinear features introduced into the pricing and hedging problems stemming from three salient features prevalent in modern day derivatives markets; nonlinear differential funding, default and collateralization. Their implications upon the arbitrage-free nature of market models and the nonlinear pricing of contingent claims is examined. A common arbitrage-free framework encompassing all three features, including an endogenous method for the determination of optimal collateral is presented in discrete-time.