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.     

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.     

Multi-agent investment in incomplete markets

The problem of the expected utility maximization in incomplete markets for a single agent is well understood in a fairly general setting. This paper studies the problem for the multi-agent case. For this case a cooperative investment game is posed as follows: firstly collect all agents capital together at the initial time, then invest the total capital in a trading strategy, and finally divide the terminal wealth of the trading strategy and each of them gets a part. We give a characterization of Pareto optimal cooperative strategies and a characterization of situations where cooperation strictly Pareto dominates non cooperation, and prove that the core of the cooperative investment game is non-empty under mild conditions using Scarf theorem.Received: August 2003, Mathematics Subject Classification (1991): 91B28, 91A12, 60H30JEL Classification: G11, C71This work is supported by the National Natural Science Foundation of China under grant 10201031. It is a pleasure for the author to express his sincere thanks to an anonymous referee for valuable suggestions.     

Optimal portfolios when stock prices follow an exponential Lévy process

We investigate some portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the risk, where we measure risk by the variance, but also by the Capital-at-Risk (CaR). The solution of the mean-variance problem has the same structure for any price process which follows an exponential Lévy process. The CaR involves a quantile of the corresponding wealth process of the portfolio. We derive a weak limit law for its approximation by a simpler Lévy process, often the sum of a drift term, a Brownian motion and a compound Poisson process. Certain relations between a Lévy process and its stochastic exponential are investigated.Received: January 2003Mathematics Subject Classification: Primary: 60F05, 60G51, 60H30, 91B28; secondary: 60E07, 91B70JEL Classification: C22, G11, D81We would like to thank Jan Kallsen and Ralf Korn for discussions and valuable remarks on a previous version of our paper. The second author would like to thank the participants of the Conference on Lévy Processes at Aarhus University in January 2002 for stimulating remarks. In particular, a discussion with Jan Rosinski on gamma processes has provided more insight into the approximation of the variance gamma model.     

Inf-convolution of risk measures and optimal risk transfer

We develop a methodology for optimal design of financial instruments aimed to hedge some forms of risk that is not traded on financial markets. The idea is to minimize the risk of the issuer under the constraint imposed by a buyer who enters the transaction if and only if her risk level remains below a given threshold. Both agents have also the opportunity to invest all their residual wealth on financial markets, but with different access to financial investments. The problem is reduced to a unique inf-convolution problem involving a transformation of the initial risk measures.Received: December 2004, Mathematics Subject Classification (2000): 60G35, 91B28, 91B30, 46N10JEL Classification: C61, D81, G13, G22     

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.     

Valuation of credit default swaps and swaptions

This paper presents a conceptual and general framework for valuation of single-name credit derivatives. The general subfiltration approach of [J-R] to modelling default risk, which includes the Cox-process setting of [L], is integrated with a numeraire invariant approach. Several known results are reformulated and extended in this framework. New concepts and results are presented for change of numeraire in presence of default and valuation of credit swaptions. A new formula on fractional recovery of pre-default value is derived, generalizing that of [D-S]. A Black-Scholes formula for credit default swaptions due to [S] is shown to serve as a least-squares approximation to the general case.Received: 1 November 2003, Mathematics Subject Classification: 91B28, 60G44, 60G40JEL Classification: E43, G13I would like to thank the editor and two anonymous referees for their valuable comments and suggestions.     

Maturity cycles in implied volatility

The skew effect in market implied volatility can be reproduced by option pricing theory based on stochastic volatility models for the price of the underlying asset. Here we study the performance of the calibration of the S&P 500 implied volatility surface using the asymptotic pricing theory under fast mean-reverting stochastic volatility described in [8]. The time-variation of the fitted skew-slope parameter shows a periodic behaviour that depends on the option maturity dates in the future, which are known in advance. By extending the mathematical analysis to incorporate model parameters which are time-varying, we show this behaviour can be explained in a manner consistent with a large model class for the underlying price dynamics with time-periodic volatility coefficients.Received: December 2003, Mathematics Subject Classification (2000): 91B70, 60F05, 60H30JEL Classification: C13, G13Jean-Pierre Fouque: Work partially supported by NSF grant DMS-0071744.Ronnie Sircar: Work supported by NSF grant DMS-0090067. We are grateful to Peter Thurston for research assistance.We thank a referee for his/her comments which improved the paper.     

On the pricing of forward starting options in Heston’s model on stochastic volatility

We consider the problem of pricing European forward starting options in the presence of stochastic volatility. By performing a change of measure using the asset price at the time of strike determination as a numeraire, we derive a closed-form solution within Hestons stochastic volatility framework applying distribution properties of the volatility process. In this paper we develop a new and more suitable formula for pricing forward starting options. This formula allows to cover the smile effects observed in a Black-Scholes environment, in which the extreme exposure of forward starting options to volatility changes is ignored.Received: July 2004, Mathematics Subject Classification (2000): 91B28, 60G44, 60H30, 60E10JEL Classification: G13It is a pleasure to thank the anonymous referee for his valuable comments and suggestions on this paper. Furthermore, we would like to thank Holger Kraft, University of Kaiserslautern, and Alexander Giese, HypoVereinsbank AG Munich, for fruitful discussions and suggestions.     

Convergence of utility functions and convergence of optimal strategies

In this paper we study the stability (in the L ^p as well as for the almost sure convergence sense) of the optimal investment-consumption strategy with respect to the choice of the utility function.Received: February 2003, The authors want to thank an anonymous referee as well as an associate editor for their useful comments and suggestions (in particular, the development on the speed of convergence). The authors also thank R. A. Dana for helpful discussions and W. Schachermayer for careful reading.     

Asymmetric information and imperfect competition in a continuous time multivariate security model

This paper deals with the problem of price formation in a market with asymmetric information and several risky assets. We then extend the multivariate security model of Caballé and Krishnan (1994) to a continuous time framework, and general utility function. Our model enables us to observe some results which are specific to multi security markets such as Giffen effect. An application of the main result will be the non trivial generalizations of the models of Back (1992) and Cho (1997).Mathematics Subject Classification (1991): 49L10, 60G44, 90A15JEL Classification: G11, G12The author would like to thank his supervisor H. Pham, K. Back and an anonymous referee for useful comments and discussions.