Topics in Optimal Stopping and Fundamental Theorem of Asset Pricing.

[摘要] In this thesis, we investigate several problems in optimal stopping and fundamental theorem of asset pricing (FTAP).In Chapter II, we study the controller-stopper problems with jumps. By a backward induction, we decompose the original problem with jumps into controller-stopper problems without jumps. Then we apply the decomposition result to indifference pricing of American options under multiple default risk.In Chapters III and IV, we consider zero-sum stopping games, where each player can adjust her own stopping strategies according to the other’s behavior. We show that the values of the games and optimal stopping strategies can be characterized by corresponding Dynkin games. We work in discrete time in Chapter III and continuous time in Chapter IV.In Chapter V, we analyze an optimal stopping problem, in which the investor can peek epsilon amount of time into the future before making her stopping decision. We characterize the solution of this problem by a path-dependent reflected backward stochastic differential equation. We also obtain the order of the value as epsilon goes to zero.In Chapters VI-VIII, we investigate arbitrage and hedging under non-dominated model uncertainty in discrete time, where stocks are traded dynamically and liquid European-style options are traded statically. In Chapter VI we obtain the FTAP and hedging dualities under some convex and closed portfolio constraints. In Chapter VII we study arbitrage and super-hedging in the case when the liquid options are quoted with bid-ask spreads. In Chapter VIII we investigate the dualities for sub and super-hedging prices of American options. Note that for these three chapters, since we work in the frameworks lacking dominating measures, many classical tools in probability theory cannot be applied.In Chapter IX, we consider arbitrage, hedging, and utility maximization in a given model, where stocks are available for dynamic trading, and both European and American options are available for static trading. Using a separating hyperplane argument, we get the result of FTAP, which implies the dualities of hedging prices. Then the hedging dualities lead to the duality for the utility maximization.

[发布机构] University of Michigan

[效力级别] stopping game

[关键词] optimal stopping;stopping game;fundamental theorem of asset pricing;hedging duality;semi-static trading strategy;Mathematics;Science;Applied and Interdisciplinary Mathematics