CONTEMPORARY MATHEMATICS

351

Mathematics of Finance Proceedings of an AMS-IMS-SIAM

Joint Summer Research Conference on Mathematics of Finance

June 22-26, 2003 Snowbird , Utah

George Yin Qing Zhang

Editors

Mathematics of Finance

http://dx.doi.org/10.1090/conm/351

CoNTEMPORARY MATHEMATICS

351

Mathematics of Finance Proceedings of an AMS-IMS-SIAM

Joint Summer Research Conference on Mathematics of Finance

June 22-26, 2003 Snowbird, Utah

George Yin Qing Zhang

Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeTurck, managing editor

Andreas Blass Andy R. Magid Michael Vogelius The 2003 AMS-IMS-SIAM Joint Summer Research Conference on "Mathematics of

Finance" was held at Snowbird, Utah, June 22-26, 2003, with support from the National Science Foundation, grant DMS 9973450.

2000 Mathematics Subject Classification. Primary 91B16, 91B26, 91B28, 60G15, 60G40, 60J10, 60J60, 93E20.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Library of Congress Cataloging-in-Publication Data AMS-IMS-SIAM Joint Summer Research Conference on Mathematics of Finance (2003 : Snow-bird, Utah)

Mathematics of finance: 2003 AMS-IMS-SIAM Joint Summer Research Conference on Math-ematics of Finance, June 22-26, 2003, Snowbird, Utah / George Yin, Qing Zhang, editors.

p. em. (Contemporary mathematics; 351) Includes bibliographical references. ISBN 0-8218-3412-6 (alk. paper) 1. Busines mathematics-Congresses. I. Yin, George, 1954- II. Zhang, Qing, 1959- III. Ti-

tle. IV. Contemporary mathematics (American Mathematical Society); v. 351.

HF569l.A63 2003 332.6101151-dc22 2004046167

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Contents

Preface ix

List of Speakers and Title of Talks xi

Credit Barrier Models in a Discrete Framework 1 CLAUDIO ALBANESE and OLIVER X. CHEN

Optimal Derivatives Design under Dynamic Risk Measures 13 PAULINE BARRIEU and NICOLE EL KAROUI

On Pricing of Forward and Futures Contracts on Zero-Coupon Bonds in the Cox-Ingersoll-Ross Model 27

J:§DRZEJ BIALKOWSKI and JACEK JAKUBOWSKI

Pricing and Hedging of Credit Risk: Replication and Mean-Variance Approaches (I) 37

TOMASZ R. BIELECKI, MONIQUE JEANBLANC, and MAREK RUTKOWSKI

Pricing and Hedging of Credit Risk: Replication and Mean-Variance Approaches (II) 55

TOMASZ R. BIELECKI, MONIQUE JEANBLANC, and MAREK RUTKOWSKI

Spot Convenience Yield Models for the Energy Markets 65 RENE CARMONA and MICHAEL LUDKOVSKI

Optimal Portfolio Management with Consumption 81 NETZAHUALCOYOTL CASTANEDA-LEYVA and DANIEL HERNANDEZ-HERNANDEZ

Some Processes Associated with a Fractional Brownian Motion 93 T. E. DUNCAN

Pricing Claims on Non Thadable Assets 103 ROBERT J. ELLIOTT and JOHN VANDER HOEK

Some Optimal Investment, Production and Consumption Models 115 WENDELL H. FLEMING

Asian Options under Multiscale Stochastic Volatility 125 JEAN-PIERRE FOUQUE and CHUAN-HSIANG HAN

A Regime Switching Model: Statistical Estimation, Empirical Evidence, and Change Point Detection 139

XIN Guo

v

vi CONTENTS

Multinomial Maximum Likelihood Estimation of Market Parameters for Stock Jump-Diffusion Models 155

FLOYD B. HANSON, JOHN J. WESTMAN, and ZONGWU ZHU

Optimal Terminal Wealth under Partial Information for HMM Stock Returns 171

ULRICH G. HAUSSMANN and JORN SASS

Computing Optimal Selling Rules for Stocks Using Linear Programming 187 KURT HELMES

Optimization of Consumption and Portfolio and Minimization of Volatility 199 YAOZHONG Hu

Options: To Buy or not to Buy? 207 MATTIAS JONSSON and RONNIE SIRCAR

Risk Sensitive Optimal Investment: Solutions of the Dynamical Programming Equation 217

H. KAISE and S. J. SHEU

Hedging Default Risk in an Incomplete Market ANDREW E.B. LIM

Mean-Variance Portfolio Choice with Discontinuous Asset Prices and Nonnegative Wealth Processes

ANDREW E.B. LIM and XUN Yu ZHOU

Indifference Prices of Early Exercise Claims MAREK MUSIELA and THALEIA ZARIPHOPOULOU

Random Walk around Some Problems in Identification and Stochastic

231

247

259

Adaptive Control with Applications to Finance 273 BOZENNA PASIK-DUNCAN

Pricing and Hedging for Incomplete Jump Diffusion Benchmark Models 287 ECKHARD PLATEN

Why is the Effect of Proportional Transaction Costs 0(8213)? 303 L.C.G. RoGERS

Estimation via Stochastic Filtering in Financial Market Models 309 WOLFGANG J. RUNGGALDIER

Stochastic Optimal Control Modeling of Debt Crises 319 JEROME L. STEIN

Duality and Risk Sensitive Portfolio Optimization 333 LUKASZ STETTNER

Characterizing Option Prices by Linear Programs 349 RICHARD H. STOCKBRIDGE

Pricing Defaultable Bond with Regime Switching 361 J.W. WANG and Q. ZHANG

CONTENTS

Affine Regime-Switching Models for Interest Rate Term Structure SHU Wu and YONG ZENG

Stochastic Approximation Methods for Some Finance Problems G. YIN and Q. ZHANG

vii

375

387

Preface

This volume contains 31 papers, based on the invited talks given at the 2003 AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences: Mathematics of Finance, held in Snowbird, Utah, June 22-26, 2003. This was the first ever conference on mathematics of finance jointly sponsored by AMS, IMS, and SIAM.

Financial mathematics is a rapidly expanding field. It involves a wide spectrum of techniques that go far beyond the traditional applied mathematics. Research in mathematics of finance has witnessed tremendous progress in recent years. The Black-Scholes model and its various extensions for pricing of options have had an influential impact on financial practice and led to a revolution in the financial in-dustry. The introduction of stochastic analysis and stochastic control techniques has resulted in a number of important advances. To name just a few, they include the studies of valuation of contingent claims in complete and incomplete markets, consumption-investment models with or without constraints, portfolio management for institutional investors such as pension funds and banks, and risk assessment and management using financial derivatives. These applications, on the other hand, re-quire and stimulate many new and exciting theoretical discoveries. As a major impetus to the development of financial management and economics, research in mathematics of finance has had a major impact on the global economy. Moreover, the development of mathematics of finance has created a large demand for mathe-matics graduates at both Master and Ph.D. levels in the financial industry, resulting in the introduction of this topic in the curriculum of mathematical sciences depart-ments of many universities. The rapid progress has necessitated communication and networking among researchers in different disciplines. This summer research conference provided us with an excellent and timely opportunity. It brought to-gether researchers from mathematical sciences, finance, economics, and engineering, and financial industry to review and to update the recent advances, and to identify future directions of mathematics of finance.

The scientific program of the conference consisted of 42 invited talks, a poster session, and a panel discussion on research and education. While recent progress has been surveyed, reviewed, and substantially updated, new ideas, models, methods, and techniques have been explored. The invited speakers presented a broad spec-trum of problems, models, and results involving modeling, estimation, optimiza-tion, control, risk assessment and management, contingent claim pricing, dynamic hedging, and financial derivative design. Valuation of contingent claims remains the centerpiece of modern financial theory. Its key components include financial market modeling and dynamic hedging. While the Black-Scholes models have been widely

ix

X PREFACE

used in characterizing movements of financial markets for decades, it has been recog-nized that their utility is limited because they assume deterministic rates of return and volatility, and because they ignore many aspects of the markets. In the past few years, various attempts (including stochastic volatility, jump diffusions, and hybrid market models) have emerged to modify and generalize the Black-Scholes models. Optimal portfolio management uses a stochastic control approach. Orig-inating from Merton's pioneering work, it continues to have an important role in finance theory. The objective is to allocate financial assets dynamically among risky and fixed-income investments with the goal of maximizing expected overall return of consumption measured by some utility function. A closed-form solution is possible only for the simplest models. Typically, optimal investment and con-sumption control policies must be found by solving a partial differential equation of Hamilton-Jacobi-Bellman type. However, nonlinearities make numerical implemen-tations difficult, and efficient schemes are needed. Other difficult issues include, for example, mathematical model selections and choices of utility functions. Financial risk management has attracted growing attention in recent years. Such devastating events as the Long-Term Capital Management default and the Enron bankruptcy shook the financial world. It has become clear that there is an urgent need for fur-ther research on corporation credit risks as well as the possibility to hedging these risks using financial derivatives. As a result, one of the emerging research topics is the study of credit risk management.

As an archive, this volume presents some of the highlights of the conference. It collects papers covering a broad spectrum of topics in mathematical finance; all pa-pers have been refereed. The organizing committee consisted of Wendell H. Fleming (Brown University), Jean-Pierre Fouque (North Carolina State University), George Papanicolaou (Stanford University), Bozenna Pasik-Duncan (University of Kansas), Stanley R. Pliska (University of Illinois at Chicago), Ronnie Sircar (Princeton Uni-versity), George Yin (Wayne State University, Chair), and Qing Zhang (University of Georgia, Co-chair). It was supported in part by the National Science Foundation.

Without the encouragement, help, and assistance of many individuals, the con-ference could not have taken place. We thank the invited speakers, the panelists, the poster presenters, and all invitees for making the conference a successful event; we thank the members of the organizing committee for their help, advice, and sug-gestions. Our thanks go to the AMS-IMS-SIAM Committee on Summer Research Conferences in the Mathematical Sciences, in particular, Thomas DiCiccio, chair of the committee, and James Maxwell of the AMS, who helped us shape the confer-ence and provided us with valuable comments and suggestions in the preparation of the conference. We are especially grateful to Donna Salter and Wayne Drady for their constant and tireless help during the preparation of the conference as on-site management. The assistance from Sergei Gelfand, Christine Thivierge, and the AMS professionals during the preparation of this volume is also gratefully acknowl-edged. Finally, we are thankful to the National Science Foundation for supporting the Summer Research Conference.

George Yin and Qing Zhang

List of Speakers and Title of Talks

(1) Claudio Albanese, Credit Barrier Models (2) Tomasz R. Bielecki, Hedging of Default Risk: Incomplete Case (3) Abel Cadenillas, Optimal Manager Compensation with Choice of Effort

and Volatility (4) Rene Carmona, Spreads, Swings, and Temperature Options: Mathemati-

cal Challenges (5) Tyrone Duncan, Some Processes Associated with a Fractional Brownian

Motion for Finance (6) Nicole El Karoui, Optimal Risks Transfer under Dynamic Risk Measure (7) Robert J. Elliott, Pricing Untradable Claims (8) Wendell H. Fleming, Some Optimal Investment, Production and Con-

sumption Models (9) Jean-Pierre Fouque, Multiscale Stochastic Volatility Asymptotics

(10) Ulrich Haussmann, Optimal Portfolios Based on Stock Prices when Rates of Return are Hidden Markov Chains

(11) Kurt Helmes, Computing Optimal Selling Rules for Stocks Using Linear Programming

(12) Yaozhong Hu, Optimal Portfolio for an Insider (13) Xin Guo, A Constrained Nonlinear Regular-Singular Control Problem

with Application (14) Floyd B. Hanson, Robust Estimation of Market Parameters for Stock

Jump-diffusion Models (15) Daniel Hermindez-Hernandez, Optimal Investment in Incomplete Finan-

cial Markets with Stochastic Volatility (16) Jacek Jakubowski, On Pricing of Futures and Forward Contracts on a

Zero Coupon Bonds in the Cox-Ingersoll-Ross Model (17) Roger Lee, Model Free Replication of Path-Dependent Securities (18) Andrew Lim, Hedging Default in an Incomplete Market (19) Jin Ma, Pathwise Stochastic Control Problems and Stochastic HJB Equa-

tions (20) Marek Musiela, Modelling in Finance: Case Study-FX Market (21) Tao Pang, A Stochastic Control Model in Risk Management (22) Bozenna Pasik-Duncan, Random Walk around Some Problems in Identi-

fication and Adaptive Control of Stochastic Systems (23) Eckhard Platen, Incomplete Benchmark Models with Intensity Based Jumps (24) Stanley R. Pliska, An Intensity-Based Approach for Valuation of Mortgage

Contracts Subject to Prepayment Risk (25) Chris Rogers, Two-Sector Stochastic Growth Models

xi

xii

(26)

(27) (28) (29) (30) (31)

(32)

(33)

(34)

(35)

(36)

(37)

(38) (39) (40)

(41) (42)

LIST OF SPEAKERS AND TITLE OF TALKS

Wolfgang Runggaldier, Estimation via Stochastic Filtering in Financial Market Models Marek Rutkowski, Hedging of Default Risk: Complete Case Ronnie Sircar, Optimal Investment with Derivatives Steven Shreve, Perpetual Convertible Bonds Jerome L. Stein, Stochastic Optimal Control Modeling of Debt Crises Lukasz Stettner, Some Financial Applications of Risk Sensitive Control Problems Srdjan Stojanovic, Reduced Monge-Ampere PDEs, Optimal Portfolio Hedg-ing, and Option Pricing Richard H. Stockbridge, Linear Programming Formulations of Singular Stochastic Control Problems Michael Taksar, Ruin Probability Minimization and Dividend Distribu-tion Optimization in Diffusion Models Hui Wang, On the Convergence from Discrete to Continuous Time in an Optimal Stopping Problem John Westman, Jump-Diffusion Model for Portfolio and Consumption Control Optimization with Dividends George Yin, Stochastic Approximation Methods for Estimation and Op-timization in Some Finance Problems Jiongmin Yong, Rolling Horizon Bond and a Discount Bond Thaleia Zariphopoulou, Valuation in Incomplete Market Environments Yong Zeng, A General Equilibrium Model of the Term Structure of Inter-est Rates under Regime-Shift Risk Qing Zhang, Hybrid Dynamic Systems and Related Financial Models Xun Yu Zhou, Continuous-Time Mean-Variance Portfolio Selection with Bankruptcy Prohibition

Titles in This Series

352 Marek Kubale, Editor, Graph colorings, 2004 351 George Yin and Qing Zhang, Editors, Mathematics of finance, 2004 350 Abbas Bahri, Sergiu Klainerman, and Michael Vogelius, Editors, Noncompact

problems at the intersection of geometry, analysis, and topology, 2004 349 Alexandre V. Borovik and Alexei G. Myasnikov, Editors, Computational and

experimental group theory, 2004 348 Hiroshi Isozaki, Editor, Inverse problems and spectral theory, 2004 347 Motoko Kotani, Tomoyuki Shirai, and Toshikazu Sunada, Editors, Discrete

geometric analysis, 2004 346 Paul Goerss and Stewart Priddy, Editors, Homotopy theory: Relations with algebraic

geometry, group cohomology, and algebraic K-theory, 2004 345 Christopher Heil, Palle E. T. Jorgensen, and David R. Larson, Editors, Wavelets,

frames and operator theory, 2004 344 Ricardo Baeza, John S. Hsia, Bill Jacob, and Alexander Prestel, Editors,

Algebraic and arithmetic theory of quadratic forms, 2004 343 N. Sthanumoorthy and Kailash C. Misra, Editors, Kac-moody lie algebras and

related topics, 2004 342 Janos Pach, Editor, Towards a theory of geometric graphs, 2004 341 Hugo Arizmendi, Carlos Bosch, and Lourdes Palacios, Editors, Topological

algebras and their applications, 2004 340 Rafael del Rio and Carlos Villegas-Blas, Editors, Spectral theory of Schrodinger

operators, 2004 339 Peter Kuchment, Editor, Waves in periodic and random media, 2003 338 Pascal Auscher, Thierry Coulhon, and Alexander Grigor'yan, Editors, Heat

kernels and analysis on manifolds, graphs, and metric spaces, 2003 337 Krishan L. Duggal and Ramesh Sharma, Editors, Recent advances in Riemannian

and Lorentzian geometries, 2003 336 Jose Gonzalez-Barrios, Jorge A. Leon, and Ana Meda, Editors, Stochastic models,

2003 335 Geoffrey L. Price, B. Mitchell Baker, Palle E.T. Jorgensen, and PaulS. Muhly,

Editors, Advances in quantum dynamics, 2003 334 Ron Goldman and Rimvydas Krasauskas, Editors, Topics in algebraic geometry and

geometric modeling, 2003 333 Giova~mi Alessandrini and Gunther Uhlmann, Editors, Inverse problems: Theory

and applications, 2003 332 John Bland, Kang-Tae Kim, and Steven G. Krantz, Editors, Explorations in

complex and Riemannian geometry, 2003 331 Luchezar L. Avramov, Marc Chardin, Marcel Morales, and Claudia Polini,

Editors, Commutative algebra: Interactions with algebraic geometry, 2003 330 S. Y. Cheng, C.-W. Shu, and T. Tang, Editors, Recent advances in scientific

computing and partial differential equations, 2003 329 Zhangxin Chen, Roland Glowinski, and Kaitai Li, Editors, Current trends in

scientific computing, 2003 328 Krzysztof Jarosz, Editor, Function spaces, 2003 327 Yulia Karpeshina, Giinter Stolz, Rudi Weikard, and Yanni Zeng, Editors,

Advances in differential equations and mathematical physics, 2003 326 Kenneth D. T-R McLaughlin and Xin Zhou, Editors, Recent developments in

integrable systems and Riemann-Hilbert problems, 2003 325 Seok-Jin Kang and Kyu~Hwan Lee, Editors, Combinatorial and geometric

representation theory, 2003

TITLES IN THIS SERIES

324 Caroline Grant Melles, Jean-Paul Brasselet, Gary Kennedy, Kristin Lauter, and Lee McEwan, Editors, Topics in algebraic and noncommutative geometry, 2003

323 Vadim Olshevsky, Editor, Fast algorithms for structured matrices: theory and applications, 2003

322 S. Dale Cutkosky, Dan Edidin, Zhenbo Qin, and Qi Zhang, Editors, Vector bundles and representation theory, 2003

321 Anna Kaminska, Editor, Trends in Banach spaces and operator theory, 2003 320 William Beckner, Alexander Nagel, Andreas Seeger, and Hart F. Smith,

Editors, Harmonic analysis at Mount Holyoke, 2003 319 W. H. Schikhof, C. Perez-Garcia, and A. Escassut, Editors, Ultrametric functional

analysis, 2003 318 David E. Radford, Fernando J. 0. Souza, and David N. Yetter, Editors,

Diagrammatic morphisms and applications, 2003 317 Hui-Hsiung Kuo and Ambar N. Sengupta, Editors, Finite and infinite dimensional

analysis in honor of Leonard Gross, 2003 316 0. Cornea, G. Lupton, J. Oprea, and D. Tanre, Editors, Lusternik-Schnirelmann

category and related topics, 2002 315 Theodore Voronov, Editor, Quantization, Poisson brackets and beyond, 2002 314 A. J. Berrick, Man Chun Leung, and Xingwang Xu, Editors, Topology and

Geometry: Commemorating SISTAG, 2002 313 M. Zuhair Nashed and Otmar Scherzer, Editors, Inverse problems, image analysis,

and medical imaging, 2002 312 Aaron Bertram, James A. Carlson, and Holger Kley, Editors, Symposium in

honor of C. H. Clemens, 2002 311 Clifford J. Earle, William J. Harvey, and Sevin Recillas-Pishmish, Editors,

Complex manifolds and hyperbolic geometry, 2002 310 Alejandro Adem, Jack Morava, and Yongbin Ruan, Editors, Orbifolds in

mathematics and physics, 2002 309 Martin Guest, Reiko Miyaoka, and Yoshihiro Ohnita, Editors, Integrable systems,

topology, and physics, 2002 308 Martin Guest, Reiko Miyaoka, and Yoshihiro Ohnita, Editors, Differentiable

geometry and integrable systems, 2002 307 Ricardo Weder, Pavel Exner, and Benoit Grebert, Editors, Mathematical results in

quantum mechanics, 2002 306 Xiaobing Feng and Tim P. Schulze, Editors, Recent advances in numerical methods

for partial differential equations and applications, 2002 305 Samuel J. Lomonaco, Jr. and Howard E. Brandt, Editors, Quantum computation

and information, 2002 304 Jorge Alberto Calvo, Kenneth C. Millett, and Eric J. Rawdon, Editors, Physical

knots: Knotting, linking, and folding geometric objects in JR3 , 2002 303 William Cherry and Chung-Chun Yang, Editors, Value distribution theory and

complex dynamics, 2002 302 Yi Zhang, Editor, Logic and algebra, 2002 301 Jerry Bona, Roy Choudhury, and David Kaup, Editors, The legacy of the inverse

scattering transform in applied mathematics, 2002

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstoref.

The mathematics of finance involves a wide spectrum of techniques that go beyond tradi-tional applied mathematics. The field has witnessed a tremendous amount of progress in recent years, which has inspired communication and networking among researchers in finance, economics, engineering, and industry. This volume contains papers based on talks given at the first AMS-IMS-SIAM Joint Summer Research Conference on Mathematics of Finance held at Snowbird (UT). Topics covered here include modeling, estimation, optimization, control, risk assessment and management, contingent claim pricing, dynamic hedging, and financial derivative design. The book is suitable for graduate students and research mathematicians interested in mathematical finance.

ISBN 0-8218-3412-6

9 780821 834121