Pricing derivatives of American and game type in incomplete markets

In this paper the neutral valuation approach is applied to American and game options in incomplete markets. Neutral prices occur if investors are utility maximizers and if derivative supply and demand are balanced. Game contingent claims are derivative contracts that can be terminated by both counterparties at any time before expiration. They generalize American options where this right is limited to the buyer of the claim. It turns out that as in the complete case, the price process of American and game contingent claims corresponds to a Snell envelope or to the value of a Dynkin game, respectively.On the technical level, an important role is played by -sub- and -supermartingales. We characterize these processes in terms of semimartingale characteristics.Received: June 2003, Mathematics Subject Classification (2000):   91B24, 60G48, 91B16, 91A15, 60G40JEL Classification:   G13, D52, C73The authors want to thank PD Dr. Martin Beibel for the idea leading to the proof of Proposition A.4 and both anonymous referees for many valuable comments. The second author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft through the Graduiertenkolleg Angewandte Algorithmische Mathematik at Munich University of Technology and by the Fonds zur Förderung der wissenschaftlichen Forschung at Vienna University of Technology.     

From the Minimal Entropy Martingale Measures to the Optimal Strategies for the Exponential Utility Maximization: the Case of Geometric Lévy Processes

In this article, we will consider a multi-dimensional geometric L'evy process as a financial market model. We will first determine the minimal entropy martingale measure (MEMM); we will next derive the optimal strategy for the exponential utility maximization of terminal wealth concretely from the representation of the MEMM. JEL Classification: D46, D52, G12 AMS (2000) Subject Classification: 60G44, 60G51, 60G52,60H20, 60J75, 91B16, 91B28, 94A17     

Hazard rate for credit risk and hedging defaultable contingent claims

We provide a concise exposition of theoretical results that appear in modeling default time as a random time, we study in details the invariance martingale property and we establish a representation theorem which leads, in a complete market setting, to the hedging portfolio of a vulnerable claim. Our main result is that, to hedge a defaultable claim one has to invest the value of this contingent claim in the defaultable zero-coupon.Received: April 2003Mathematics Subject Classification: 91B24, 91B29, 60G46JEL Classification: G10The authors would like to thank D. Becherer and J.N. Hugonnier for interesting discussions and the anonymous referee whose pertinent questions on the first version of this paper help them to clarify the proofs. All remaining errors are ours.     

Robust utility maximization for complete and incomplete market models

We investigate the problem of maximizing the robust utility functional . We give the dual characterization for its solution for both a complete and an incomplete market model. To this end, we introduce the new notion of reverse f-projections and use techniques developed for f-divergences. This is a suitable tool to reduce the robust problem to the classical problem of utility maximization under a certain measure: the reverse f-projection. Furthermore, we give the dual characterization for a closely related problem, the minimization of expenditures given a minimum level of expected utility in a robust setting and for an incomplete market.Received: September 2004, Mathematics Subject Classification (2000): 62C20, 62O05, 91B16, 91B28JEL Classification: D81, G11I thank Hans Föllmer for his help when writing this paper. Furthermore, I thank Alexander Schied for discussing the topic with me and Michael Kupper and the referees for their helpful remarks.