Utility Based Pricing and Hedging in Incomplete Markets with Stochastic Preferences in a Unifying Framework of Admissibility

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The problems of pricing and hedging of financial instruments are of fundamental importance from both theoretical and practical sides of mathematical finance. In this research project, the investigator develops a novel framework and an approach for analysis of various notions of prices and hedging strategies in general models of financial markets. He aims to develop a utility-based pricing method for contingent claims with embedded payment streams. In a complete financial market every contingent claim can be replicated by a portfolio of the traded securities and therefore admits a unique arbitrage-free price, which is an initial value of the replicating strategy. In an incomplete market, to every contingent claim is associated an interval of arbitrage-free prices (unless the claim is replicable). In order to overcome the issue of non-uniqueness, alternative (equilibrium- or utility-based) approaches have been developed. The present project focuses on the probabilistic process-theoretic aspects of the problems of utility-based pricing and hedging of contingent claims and income streams. The aim of this project is twofold -- investigation of different notions of prices, hedging strategies, and relationships between them, in the settings that include stochastic utility (as a natural generalization of the utility function concept), and a construction of a utility-based pricing methodology for the contingent claims with embedded payment streams, in incomplete continuous-time models of financial markets. A novel concept of a unified framework of admissibility is introduced for the accomplishment of the goals described above. The unified framework permits consideration of the underlying utility maximization problems with different types of budget constraints, in one formulation. On the technical level, the analysis relies on the existing and new techniques drawing on the general theory of semimartingales, stochastic optimal control, and convex analysis. The approach provides a firm ground for an extension of the scope of applicability of the utility-based pricing techniques and for asymptotic expansions of the utility-based prices and hedging strategies.