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Springer Texts in Statistics Series Editors: G. Casella S. Fienberg I. Olkin

Springer Texts in Statistics For other titles published in this series, go to www.springer.com/series/417

Tze Leung Lai · Haipeng Xing Statistical Models and Methods for Financial Markets 123

Tze Leung Lai Department of Statistics Stanford University Stanford, CA 94305 USA lait@stanford.edu Haipeng Xing Department of Statistics Columbia University New York, NY 10027 USA xing@stat.columbia.edu Editorial Board George Casella Department of Statistics University of Florida Gainesville, FL 32611-8545 USA Stephen Fienberg Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213-3890 USA Ingram Okin Department of Statistics Stanford University Stanford, CA 94305 USA ISBN: 978-0-387-77826-6 DOI: 10.1007/978-0-387-77827-3 e-ISBN: 978-0-387-77827-3 Library of Congress Control Number: 2008930111 c 2008 Springer Science+Business Media, LLC 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, LLC, 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 known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they 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 on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

To Letitia and Ying

Preface The idea of writing this book arose in 2000 when the ﬁrst author was assigned to teach the required course STATS 240 (Statistical Methods in Finance) in the new M.S. program in ﬁnancial mathematics at Stanford, which is an interdisciplinary program that aims to provide a master’s-level education in applied mathematics, statistics, computing, ﬁnance, and economics. Students in the program had diﬀerent backgrounds in statistics. Some had only taken a basic course in statistical inference, while others had taken a broad spectrum of M.S.- and Ph.D.-level statistics courses. On the other hand, all of them had already taken required core courses in investment theory and derivative pricing, and STATS 240 was supposed to link the theory and pricing formulas to real-world data and pricing or investment strategies. Besides students in the program, the course also attracted many students from other departments in the university, further increasing the heterogeneity of students, as many of them had a strong background in mathematical and statistical modeling from the mathematical, physical, and engineering sciences but no previous experience in ﬁnance. To address the diversity in background but common strong interest in the subject and in a potential career as a “quant” in the ﬁnancial industry, the course material was carefully chosen not only to present basic statistical methods of importance to quantitative ﬁnance but also to summarize domain knowledge in ﬁnance and show how it can be combined with statistical modeling in ﬁnancial analysis and decision making. The course material evolved over the years, especially after the second author helped as the head TA during the years 2004 and 2005. The course also expanded to include a section oﬀered by the Stanford Center for Profes- sional Development, with nondegree students in the ﬁnancial industry taking it on-line (http://scpd.stanford.edu). The steady increase in both student interest and course material led to splitting the single course into two in 2006, with STATS 240 followed by STATS 241 (Statistical Modeling in Financial Markets). Part I of this book, Basic Statistical Methods and Financial

viii Preface Applications, is covered in STATS 240 and has six chapters. Chapters 1 and 2 cover linear regression, multivariate analysis, and maximum likelihood. These statistical methods are applied in Chapter 3 to a fundamental topic in quantitative ﬁnance, namely portfolio theory and investment models, for which Harry Markowitz and William Sharpe were awarded Nobel Prizes in Economics. Whereas the theory assumes the model parameters are known, in practice the parameters have to be estimated from historical data, and Chapter 3 addresses the statistical issues and describes various statistical ap- proaches. One approach is deferred to Section 4.4 in Chapter 4, where we introduce Bayesian methods after further discussion of likelihood inference for parametric models and its applications to logistic regression and other generalized linear models that extend the linear regression models of Chapter 1 via certain “link functions.” Chapter 4 also extends the least squares method in Chapter 1 to nonlinear regression models. This provides the background for the nonlinear least squares approach, which is used in various places in Part II of the book. Another important topic in quantitative ﬁnance that has attracted considerable attention in recent years, especially after the 2003 Nobel Prizes to Robert Engle and Clive Granger, is ﬁnancial time series. Af- ter introducing the basic ideas and models in time series analysis in Chapter 5, Chapter 6 extends them to develop dynamic models of asset returns and their volatilities. The six chapters that constitute Part I of the book provide students in ﬁnancial mathematics (or engineering) and mathematical (or com- putational) ﬁnance programs with basic training in statistics in a single course that also covers two fundamental topics in quantitative ﬁnance to illustrate the relevance and applications of statistical methods. Part II of the book, Advanced Topics in Quantitative Finance, is covered in STATS 241 at Stanford. It introduces nonparametric regression in Chapter 7, which also applies the methodology to develop a substantive-empirical ap- proach that combines domain knowledge (economic theory and market prac- tice) with statistical modeling (via nonparametric regression). This approach provides a systematic and versatile tool to link the theory and formulas stu- dents have learned in mathematical ﬁnance courses to market data. A case in point is option pricing theory, the importance of which in ﬁnancial economics led to Nobel Prizes for Robert Merton and Myron Scholes in 1997, and which is a basic topic taught in introductory mathematical ﬁnance courses. Discrepan- cies between the theoretical and observed option prices are revealed in certain patterns of “implied volatilities,” and their statistical properties are studied in Chapter 8. Section 8.3 describes several approaches in the literature to ad- dress these discrepancies and considers in particular a substantive-empirical approach having a substantive component associated with the classical Black- Scholes formula and an empirical component that uses nonparametric regres- sion to model market deviations from the Black-Scholes formula. Chapter 9 introduces advanced multivariate and time series methods in ﬁnancial econo-

Preface ix metrics. It provides several important tools for analyzing time series data on interest rates with diﬀerent maturities in Chapter 10, which also relates the statistical (empirical) analysis of real-world interest rates to stochastic process models for the valuation of interest rate derivatives in mathematical ﬁnance. The ﬁnance theories in Chapters 8 and 10 and in mathematical ﬁnance courses assume the absence of arbitrage. “Statistical arbitrage,” which has become an important activity of many hedge fund managers, uses statistical learning from market prices and trading patterns to identify arbitrage opportunities in ﬁnancial markets. Chapter 11 considers statistical trading strategies and related topics such as market microstructure, data-snooping checks, transac- tion costs, and dynamic trading. Chapter 12 applies the statistical methods in previous chapters and also describes new ones for risk management from the corporate/regulatory perspective, protecting the ﬁnancial institution and its investors in case rare adverse events occur. Since M.S. students in ﬁnancial mathematics/engineering programs have strong mathematical backgrounds, the mathematical level of the book is tar- geted toward this audience. On the other hand, the book does not assume many prerequisites in statistics and ﬁnance. It attempts to develop the key methods and concepts, and their interrelationships, in a self-contained man- ner. It is also intended for analysts in the ﬁnancial industry, as mentioned above in connection with the Stanford Center for Professional Development, and exposes them to modern advances in statistics by relating these advances directly to ﬁnancial modeling. Because Part I is intended for a one-semester (or one-quarter) course, it is presented in a focused and concise manner to help students study and re- view the material for exercises, projects, and examinations. The instructor can provide more detailed explanations of the major ideas and motivational ex- amples in lectures, while teaching assistants can help those students who lack the background by giving tutorials and discussion sessions. Our experience has been that this system works well with the book. Another method that we have used to help students master this relatively large amount of material in a single course is to assign team projects so that they can learn together and from each other. Besides students in the M.S. program in ﬁnancial mathematics at Stanford, the course STATS 241, which uses Part II of the book, has also attracted Ph.D. students in economics, engineering, mathematics, statistics, and the Graduate School of Business. It attempts to prepare students for quantitative ﬁnancial research in industry and academia. Because of the breadth and depth of the topics covered in Part II of the book, the course contents actually vary from year to year, focusing on certain topics in one year and other topics in the next. Again, team projects have proved useful for students to learn together and from each other.

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