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.     

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.     

In which financial markets do mutual fund theorems hold true?

The mutual fund theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T, where agents maximize expected utility of terminal wealth. The main results are:

Wealth-path dependent utility maximization in incomplete markets

Motivated by an optimal investment problem under time horizon uncertainty and when default may occur, we study a general structure for an incomplete semimartingale model extending the classical terminal wealth utility maximization problem. This modelling leads to the formulation of a wealth-path dependent utility maximization problem. Our main result is an extension of the well-known dual formulation to this context. In contrast with the usual duality approach, we work directly on the primal problem. Sufficient conditions for characterizing the optimal solution are also provided in the case of complete markets, and are illustrated by examples.Received: December 2003, Mathematics Subject Classification (2000): 91B28, 91B16, 49N15, 49N30JEL Classification: G11The authors would like to thank the anonymous referees for their remarks and suggestions which greatly improved this paper. We also thank participants at the Oberwolfach workshop in 2003 for comments and discussions.     

A geometric approach to portfolio optimization in models with transaction costs

We consider a continuous-time stochastic optimization problem with infinite horizon, linear dynamics, and cone constraints which includes as a particular case portfolio selection problems under transaction costs for models of stock and currency markets. Using an appropriate geometric formalism we show that the Bellman function is the unique viscosity solution of a HJB equation.Mathematics Subject Classification (1991): 60G44JEL Classification: G13, G11This research was done at Munich University of Technology supported by a Mercator Guest Professorship of the German Science Foundation (Deutsche Forschungsgemeinschaft). The authors also express their thanks to Mark Davis, Steve Shreve, and Michael Taksar for useful discussions concerning the principle of dynamic programming.     

Lookback options and diffusion hitting times: A spectral expansion approach

Lookback options have payoffs dependent on the maximum and/or minimum of the underlying price attained during the options lifetime. Based on the relationship between diffusion maximum and minimum and hitting times and the spectral decomposition of diffusion hitting times, this paper gives an analytical characterization of lookback option prices in terms of spectral expansions. In particular, analytical solutions for lookback options under the constant elasticity of variance (CEV) diffusion are obtained.Received: 1 October 2003, Mathematics Subject Classification: 60J35, 60J60, 60G70JEL Classification: G13The author thanks Phelim Boyle for bringing the problem of pricing lookback options under the CEV process to his attention and for useful discussions and Viatcheslav Gorovoi for computational assistance. This research was supported by the U.S. National Science Foundation under grants DMI-0200429 and DMS-0223354.     

Arbitrage Opportunities in Diverse Markets via a Non-equivalent Measure Change

We study arbitrage opportunities in diverse markets as introduced by Fernholz (J Math Econ 31:393–417, 1999). By a change of measure technique we are able to generate a variety of diverse markets. The construction is based on an absolutely continuous but non-equivalent measure change which implies the existence of instantaneous arbitrage opportunities in diverse markets. For this technique to work, we single out a crucial non-degeneracy condition. Moreover, we discuss the dynamics of the price process under the new measure as well as further applications.Both authors gratefully acknowledge financial support by the National Centre of Competence in Research “Financial Valuation and Risk Management” (NCCR FINRISK). The NCCR FINRISK is a research program supported by the Swiss National Science Foundation. The first author also thanks Credit Suisse Group for financial support.     

Satisfying convex risk limits by trading

A random variable, representing the final position of a trading strategy, is deemed acceptable if under each of a variety of probability measures its expectation dominates a floor associated with the measure. The set of random variables representing pre-final positions from which it is possible to trade to final acceptability is characterized. In particular, the set of initial capitals from which one can trade to final acceptability is shown to be a closed half-line . Methods for computing are provided, and the application of these ideas to derivative security pricing is developed.Received: May 2004, Mathematics Subject Classification (2000): 91B30, 60H30, 60G44JEL Classification: G10Steven E. Shreve: Work supported by the National Science Foundation under grants DMS-0103814 and DMS-0139911.Reha Tütüncü: Work supported by National Science Foundation under grants CCR-9875559 and DMS-0139911.