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«Essays and Surveys in Global Optimization Charles Audet, Pierre Hansen, and Gilles Savard, editors Graph Theory and Combinatorial Optimization David ...»

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NUMERICAL METHODS IN FINANCE

GERAD 25th Anniversary Series

Essays and Surveys in Global Optimization

Charles Audet, Pierre Hansen, and Gilles Savard, editors

Graph Theory and Combinatorial Optimization

David Avis, Alain Hertz, and Odile Marcotte, editors

Numerical Methods in Finance

Hatem Ben-Ameur and Michele Breton, editors

Analysis, Control and Optimization of Complex Dynamic Sys-

tems

El-Kebir Boukas and Roland Malhame, editors

Column Generation Guy Desaulniers, Jacques Desrosiers, and Marius M. Solomon, editors Statistical Modeling and Analysis for Complex Data Problems Pierre Duchesne and Bruno Remillard, editors Performance Evaluation and Planning Methods for the Next Generation Internet Andre Girard, Brunilde Sanso, and Felisa Vazquez-Abad, editors Dynamic Games: Theory and Applications Alain Haurie and Georges Zaccour, editors Logistics Systems: Design and Optimization Andre Langevin and Diane Riopel, editors Energy and Environment Richard Loulou, Jean-Philippe Waaub, and Georges Zaccour, editors

NUMERICAL METHODS IN FINANCE

Edited by

MICHELE BRETON

GERAD and HEC Montreal

HATEM BEN-AMEUR

GERAD and HEC Montreal 4y Springer ISBN-10: 0-387-25117-0 (HB) ISBN-10: 0-387-25118-9 (e-book) ISBN-13: 978-0387-25117-2 (HB) ISBN-13: 978-0387-25118-9 (e-book) © 2005 by Springer Science+Business Media, Inc.

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science + Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed in the United States of America.

987654321 SPIN 11305408 springeronline.com Foreword GERAD celebrates this year its 25th anniversary. The Center was created in 1980 by a small group of professors and researchers of HEC Montreal, McGill University and of the Ecole Polytechnique de Montreal.

GERAD's activities achieved sufficient scope to justify its conversion in June 1988 into a Joint Research Centre of HEC Montreal, the Ecole Polytechnique de Montreal and McGill University. In 1996, the Uni- versite du Quebec a Montreal joined these three institutions. GERAD has fifty members (professors), more than twenty research associates and post doctoral students and more than two hundreds master and Ph.D.

students.

GERAD is a multi-university center and a vital forum for the develop- ment of operations research. Its mission is defined around the following

four complementarily objectives:

• The original and expert contribution to all research fields in GERAD's area of expertise;

• The dissemination of research results in the best scientific outlets as well as in the society in general;

• The training of graduate students and post doctoral researchers;

• The contribution to the economic community by solving important problems and providing transferable tools.

GERAD's research thrusts and fields of expertise are as follows:

• Development of mathematical analysis tools and techniques to solve the complex problems that arise in management sciences and engineering;

• Development of algorithms to resolve such problems efficiently;

• Application of these techniques and tools to problems posed in related disciplines, such as statistics, financial engineering, game theory and artificial intelligence;

• Application of advanced tools to optimization and planning of large technical and economic systems, such as energy systems, transportation/communication networks, and production systems;

• Integration of scientific findings into software, expert systems and decision-support systems that can be used by industry.

One of the marking events of the celebrations of the 25th anniversary of GERAD is the publication of ten volumes covering most of the Center's research areas of expertise. The list follows: Essays and Surveys in Global Optimization, edited by C. Audet, P. Hansen and G. Savard; Graph Theory and Combinatorial Optimization,

vi NUMERICAL METHODS IN FINANCE

edited by D. Avis, A. Hertz and O. Marcotte; Numerical Methods in Finance, edited by H. Ben-Ameur and M. Breton; Analysis, Control and Optimization of Complex Dynamic Systems, edited by E.K. Boukas and R. Malhame; Column Generation, edited by G. Desaulniers, J. Desrosiers and M.M. Solomon; Statistical Modeling and Analysis for Complex Data Problems, edited by P. Duchesne and B. Remillard; Performance Evaluation and Planning Methods for the Next Generation Internet, edited by A. Girard, B. Sanso and F. Vazquez-Abad; Dynamic Games: Theory and Applications, edited by A. Haurie and G. Zaccour; Logistics Systems: Design and Optimization, edited by A. Langevin and D. Riopel; Energy and Environment, edited by R. Loulou, J.-P. Waaub and G. Zaccour.





I would like to express my gratitude to the Editors of the ten volumes, to the authors who accepted with great enthusiasm to submit their work and to the reviewers for their benevolent work and timely response.

I would also like to thank Mrs. Nicole Paradis, Francine Benoit and Louise Letendre and Mr. Andre Montpetit for their excellent editing work.

The GERAD group has earned its reputation as a worldwide leader in its field. This is certainly due to the enthusiasm and motivation of GERAD's researchers and students, but also to the funding and the infrastructures available. I would like to seize the opportunity to thank the organizations that, from the beginning, believed in the potential and the value of GERAD and have supported it over the years. These are HEC Montreal, Ecole Polytechnique de Montreal, McGill University, Universite du Quebec a Montreal and, of course, the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds quebecois de la recherche sur la nature et les technologies (FQRNT).

Georges Zaccour Director of GERAD Avant-propos Le Groupe d'etudes et de recherche en analyse des decisions (GERAD) fete cette annee son vmgt-cinquieme anniversaire. Fonde en 1980 par une poignee de professeurs et chercheurs de HEC Montreal engages dans des recherches en equipe avec des collegues de l'Universite McGill et de l'Ecole Polytechnique de Montreal, le Centre comporte maintenant une cinquantaine de membres, plus d'une vingtaine de professionnels de recherche et stagiaires post-doctoraux et plus de 200 etudiants des cycles superieurs. Les activites du GERAD ont pris suffisamment d'ampleur pour justifier en juin 1988 sa transformation en un Centre de recherche conjoint de HEC Montreal, de l'Ecole Polytechnique de Montreal et de l'Universite McGill. En 1996, l'Universite du Quebec a Montreal s'est jointe a ces institutions pour parrainer le GERAD.

Le GERAD est un regroupement de chercheurs autour de la discipline de la recherche operationnelle. Sa mission s'articule autour des objectifs

complement air es suivants :

• la contribution originate et experte dans tous les axes de recherche de ses champs de competence;

• la diffusion des resultats dans les plus grandes revues du domaine ainsi qu'aupres des differents publics qui forment l'environnement du Centre;

• la formation d'etudiants des cycles superieurs et de stagiaires postdoctoraux ;

• la contribution a la communaute economique a travers la resolution de problemes et le developpement de coffres d'outils transferables.

Les principaux axes de recherche du GERAD, en allant du plus theorique au plus applique, sont les suivants :

• le developpement d'outils et de techniques d'analyse mathematiques de la recherche operationnelle pour la resolution de problemes complexes qui se posent dans les sciences de la gestion et du genie;

• la confection d'algorithmes permettant la resolution efficace de ces problemes;

• 1'application de ces outils a des problemes poses dans des disciplines connexes a la recherche operationnelle telles que la statistique, l'ingenierie financiere, la theorie des jeux et l'intelligence artificielle;

• l'application de ces outils a l'optimisation et a la planification de grands systemes technico-economiques comme les systemes energetiques, les reseaux de telecommunication et de transport, la logistique et la distributique dans les industries manufacturieres et de service;

viii NUMERICAL METHODS IN FINANCE

• l'integration des resultats scientifiques dans des logiciels, des systemes experts et dans des systemes d'aide a la decision transferables a Pindustrie.

Le fait marquant des celebrations du 25e du GERAD est la publication

de dix volumes couvrant les champs d'expertise du Centre. La liste suit :

Essays and Surveys in Global Optimization, edite par C. Audet, P. Hansen et G. Savard; Graph Theory and Combinatorial Optimization, edite par D. Avis, A. Hertz et O. Marcotte; Numerical Methods in Finance, edite par H. Ben-Ameur et M. Breton; Analysis, Control and Optimization of Complex Dynamic Systems, edite par E.K. Boukas et R. Malhame; Column Generation, edite par G. Desaulniers, J. Desrosiers et M.M. Solomon; Statistical Modeling and Analysis for Complex Data Problems, edite par P. Duchesne et B. Remillard; Performance Evaluation and Planning Methods for the Next Generation Internet, edite par A. Girard, B. Sanso et F. Vazquez-Abad; Dynamic Games : Theory and Applications, edite par A. Haurie et G. Zaccour; Logistics Systems : Design and Optimization, edite par A. Langevin et D. Riopel; Energy and Environment, edite par R. Loulou, J.-P. Waaub et G. Zaccour.

Je voudrais remercier tres sincerement les editeurs de ces volumes, les nombreux auteurs qui ont tres volontiers repondu a l'invitation des editeurs a soumettre leurs travaux, et les evaluateurs pour leur benevolat et ponctualite. Je voudrais aussi remercier Mmes Nicole Paradis, Francine Benoit et Louise Letendre ainsi que M. Andre Mont pet it pour leur travail expert d'edition.

La place de premier plan qu'occupe le GERAD sur Pechiquier mondial est certes due a la passion qui anime ses chercheurs et ses etudiants, mais aussi au financement et a Infrastructure disponibles. Je voudrais profiter de cette occasion pour remercier les organisations qui ont cru des le depart au potentiel et la valeur du GERAD et nous ont soutenus durant ces annees. II s'agit de HEC Montreal, PEcole Polytechnique de Montreal, PUniversite McGill, PUniversite du Quebec a Montreal et, bien sur, le Conseil de recherche en sciences naturelles et en genie du Canada (CRSNG) et le Fonds quebecois de la recherche sur la nature et les technologies (FQRNT).

–  –  –

The Robust Control Approach to Option Pricing and Interval 91 Models: An Overview P. Bernhard 5 A Finite Element Method for Two Factor Convertible Bonds 109 J. de Frutos 6 On Numerical Methods and the Valuation of American Options 129 M. Bellalah 7 Valuing American Contingent Claims when Time to Maturity is 143 Uncertain T. Berrada Foreign Direct Investment: The Incentive to Expropriate and the 159 Cost of Expropriation Risk E. Clark 9 Exact Multivariate Tests of Asset Pricing Models with Stable 173 Asymmetric Distributions M.-C. Beaulieu, J.-M. Dujour, and L. Khalaf

x NUMERICAL METHODS IN FINANCE

10 A Stochastic Discount Factor-Based Approach for Fixed-income 193 Mutual Fund Performance Evaluation M.A. Ayadi and L. Kryzanowski 11 Portfolio Selection with Skewness 227 P. Boyle and B. Ding 12 Continuous Min-Max Approach for Single Period Portfolio 241 Selection Problem N. Giilpinar and B. Rustem Contributing Authors

JEAN-PIERRE AUBIN DANIEL DUFRESNE

Universite Paris-Dauphine, Prance University of Melbourne, Australia j ean-pierre. aubin@viab. dauphine. f r duf resneOunimelb. edu. au

–  –  –

This volume collects twelve chapters dealing with a wide range of topics in numerical finance. It is divided in three parts. The first part contains surveys and tutorial contributions, reviewing the current state of the art in diverse mathematical methods and models and their applications in finance. The second part examines asset pricing, proposing numerical methods and specific applications. Finally, the third part deals with asset portfolios, presenting methods for efficiency testing, performance evaluation and optimal selection, with empirical experiments.

Part I In Chapter 1, P. Frangois presents a survey of the major models of the structural approach for the valuation of corporate debt in a continuoustime arbitrage-free economy. Numerous models are presented, including endogenous capital structure, discrete coupon payments, flow-based state variables, interest rate risk, strategic debt service and advanced default rules. Finally, the author presents an assessment of the performance of these structural models in capturing the empirical patterns of the term structure of credit spread.

Chapter 2 prepared by D. Dufresne is a concise account of the connection between Bessel processes and the integral of geometric Brownian motion. The main motivation for the study of this integral is the pricing of Asian options. The author reviews the definition and properties of Bessel processes. The expressions for the density function of the integral and of the Laplace transform for Asian option prices are given.

Some new derivations and alternative proofs for these results are also presented.



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