Editorial

Original Papers

Editorial     

On the use of measure-valued strategies in bond markets

We propose here a theory of cylindrical stochastic integration, recently developed by Mikulevicius and Rozovskii, as mathematical background to the theory of bond markets. In this theory, since there is a continuum of securities, it seems natural to define a portfolio as a measure on maturities. However, it turns out that this set of strategies is not complete, and the theory of cylindrical integration allows one to overcome this difficulty. Our approach generalizes the measure-valued strategies: this explains some known results, such as approximate completeness, but at the same time it also shows that either the optimal strategy is based on a finite number of bonds or it is not necessarily a measure-valued process.Received: November 2002, Mathematics Subject Classification: 60H05, 60G60, 90A09JEL Classification: G10, E43The first author gratefully acknowledges financial support from the CNR Strategic Project Modellizzazione matematica di fenomeni economici. We thank professors A. Bagchi, R. Douady and J. Zabczyk for helpful discussions. A special thanks goes to professors T. Björk, Y. Kabanov and W. Schachermayer for comments and suggestions which contributed to improve the final version of this paper.     

Liquidity risk and arbitrage pricing theory

Classical theories of financial markets assume an infinitely liquid market and that all traders act as price takers. This theory is a good approximation for highly liquid stocks, although even there it does not apply well for large traders or for modelling transaction costs. We extend the classical approach by formulating a new model that takes into account illiquidities. Our approach hypothesizes a stochastic supply curve for a securitys price as a function of trade size. This leads to a new definition of a self-financing trading strategy, additional restrictions on hedging strategies, and some interesting mathematical issues.Received: 1 November 2003, Mathematics Subject Classification: 60G44, 60H05, 90A09JEL Classification: G11, G12, G13Umut Çetin: This work was performed while Dr. Çetin was at the Center for Applied Mathematics, Cornell UniversityPhilip Protter: Supported in part by NSF grant DMS-0202958 and NSA grant MDA-904-03-1-0092 The authors wish to thank M. Warachka and Kiseop Lee for helpful comments, as well as the anonymous referee and Associate Editor for numerous helpful suggestions, which have made this a much improved paper.     

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.     

Asymptotic analysis for optimal investment and consumption with transaction costs

We consider an agent who invests in a stock and a money market and consumes in order to maximize the utility of consumption over an infinite planning horizon in the presence of a proportional transaction cost . The utility function is of the form U(c) = c^1-p/(1-p) for p > 0, . We provide a heuristic and a rigorous derivation of the asymptotic expansion of the value function in powers of , and we also obtain asymptotic results on the boundary of the no-trade region.Received: July 2003, Mathematics Subject Classification (1991): 90A09, 60H30, 60G44JEL Classification: G13Work supported by the National Science Foundation under grants DMS-0103814 and DMS-0139911.     

Large portfolio losses

This paper provide a large-deviations approximation of the tail distribution of total financial losses on a portfolio consisting of many positions. Applications include the total default losses on a bank portfolio, or the total claims against an insurer. The results may be useful in allocating exposure limits, and in allocating risk capital across different lines of business. Assuming that, for a given total loss, the distress caused by the loss is larger if the loss occurs within a smaller time period, we provide a large-deviations estimate of the likelihood that there will exist a sub-period of the future planning period during which a total loss of the critical severity occurs. Under conditions, this calculation is reduced to the calculation of the likelihood of the same sized loss over an initial time interval whose length is a property of the portfolio and the critical loss level.Received: March 2003Mathematics Subject Classification: 60F10, 91B28, 91B28JEL Classification: G21, G22, G33Amir Dembo is with the Department of Statistics, Stanford University. His research was partially supported by NSF grant #DMS-0072331. Jean-Dominique Deuschel is with the Department of Mathematics, Technische Universität, Berlin. His research was partially supported by DFG grant #663/2-3 and DFG FZT 86. Darrell Duffie is with the Graduate School of Business, Stanford University. We are extremely grateful for research assistance by Nicolae Gârleanu and Gustavo Manso, for conversations with Michael Gordy, and for comments from Michael Stutzer, Peter Carr, David Heath, and David Siegmund.     

The Disciplining Role of Accounting in the Long-Run

One role of accounting is to discipline softer (more manipulable) sources of information. We use a principal-agent model of hidden actions and hidden information to study this role. In our model, there is both a verifiable signal (a publicly observed output) and an unverifiable signal (a productivity parameter privately observed by the agent). In a one-period setting, the optimal contract does not make use of the agents report on the private signal. However, when the output is tracked over two periods, the agents communication can be valuable. This reversal of results suggests uncovering the disciplining role of accounting may require a long-term perspective.JEL Classification: D82, M41     

The Year in Review: Economics at the Antitrust Division 2003–2004

This paper covers the activities of the Economic Analysis Group (EAG), during 2003–2004. It describes the economic analysis undertaken by EAG in several important investigations, litigations, and administrative and appellate matters.The authors are, respectively, Deputy Assistant Attorney General for Economics, U.S. Department of Justice, Antitrust Division, and Economics Director of Enforcement, U.S. Department of Justice, Antitrust Division. The views contained herein are solely those of the authors and do not necessarily represent those of the U.S. Department of Justice.