Lecture Notes in Economics and Mathematical Systems 506 Founding Editors: M. Beckmann H. P. Kiinzi Managing Editors: Prof. Dr. G. Fandel Fachbereich Wirtschaftswissenschaften Femuniversitat Hagen Feithstr. 140/AVZ II, 58084 Hagen, Germany Prof. Dr. W. Trockel Institut fUr Mathematische Wirtschaftsforschung (IMW) Universitat Bielefeld Universitatsstr. 25, 33615 Bielefeld, Germany Co-Editors: C. D. Aliprantis, Dan Kovenock Editorial Board: P. Bardsley, A. Basile, M.R. Baye, T. Cason, R. Deneckere, A. Drexl, G. Feichtinger, M. Florenzano, W Giith, K. Inderfurth, M. Kaneko, P. Korhonen, W. Kiirsten, M. Li Calzi, P. K. Monteiro, Ch. Noussair, G. Philips, U. Schittko, P. Schonfeld, R. Selten, G. Sorger, R. Steuer, F. Vega-Redondo, A. P. Villamil, M. Wooders Springer-Verlag Berlin Heidelberg GmbH B. Philipp Kellerhals Financial Pricing Models in Continuous Time and Kalman Filtering Springer Author Dr. B. Philipp Kellerhals Deutscher Investment -Trust Gesellschaft fUr Wertpapieranlagen mbH Mainzer LandstraBe 11-13 60329 Frankfurt am Main, Germany Cataloging-in-Publication data applied for Die Deutsche Bibliothek -CIP-Einheitsaufnahme Kellerhals, Philipp B.: Financial pricing models in continuous time and Kalman filtering 1 B. Philipp Kellerhals. -Berlin; Heidelberg; New York; Barcelona; Hong Kong ; London; Milan; Paris; Singapore; Tokyo: Springer, 2001 (Lecture notes in economics and mathematical systems; 506) ISBN 3-540-42364-8 ISSN 0075-8450 ISBN 978-3-540-42364-5 ISBN 978-3-662-21901-0 (eBook) DOI 10.1007/978-3-662-21901-0 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. http://www.springer.de © Springer-Verlag Berlin Heidelberg 2001 Originally published by Springer-Verlag Berlin Heidelberg New York in 2001. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera ready by author/editors Cover design: design & production, Heidelberg Printed on acid-free paper SPIN: 10844260 55/3142/du 5432 I 0 To my parents Foreword Straight after its invention in the early sixties, the Kalman filter approach became part of the astronautical guidance system of the Apollo project and therefore received immediate acceptance in the field of electrical engineer ing. This sounds similar to the well known success story of the Black-Scholes model in finance, which has been implemented by the Chicago Board of Op tions Exchange (CBOE) within a few month after its publication in 1973. Recently, the Kalman filter approach has been discovered as a comfortable estimation tool in continuous time finance, bringing together seemingly un related methods from different fields. Dr. B. Philipp Kellerhals contributes to this topic in several respects. Specialized versions of the Kalman filter are developed and implemented for three different continuous time pricing models: A pricing model for closed-end funds, taking advantage from the fact, that the net asset value is observable, a term structure model, where the market price of risk itself is a stochastic variable, and a model for electricity forwards, where the volatility of the price process is stochastic. Beside the fact that these three models can be treated independently, the book as a whole gives the interested reader a comprehensive account of the requirements and capabilities of the Kalman filter applied to finance models. While the first model uses a linear version of the filter, the second model using LIBOR and swap market data requires an extended Kalman filter. Finally, the third model leads to a non-linear transition equation of the filter algorithm. Having some share in the design of the first two models, I am rather impressed by the potential and results of the filtering approach presented by Dr. Kellerhals and I hope, that this monograph will play its part in making Kalman filtering even more popular in finance. Tiibingen, May 2001 Rainer SchObel Acknowledgements The book on hand is based on my Ph.D. thesis submitted to and accepted by the College of Economics and Business Administration at the Eberhard Karls-University TUbingen, Germany. There, I had the opportunity to work as a researcher and lecturer in the Department of Finance. First and foremost, I would like to thank my academic supervisor and teacher Prof. Dr.-Ing. Rainer Schobel who introduced me to the challenging field of continuous time modeling in financial economics. He has created an ideal environment for productive research in which I have received valuable advice and support. Moreover, I am grateful to the further members of my thesis committee Prof. Dr. Werner Neus, Prof. Dr. Gerd Ronning, and Prof. Dr. Manfred Stadler. The first remarkable contact with financial theory I gratefully received during my academic year I could spend at Arizona State University, Tempe. During my final graduate studies at the University of Mannheim, Ger many, Prof. Dr. Wolfgang Buhler and Dr. Marliese Uhrig-Homburg started to create my affinity to interest rate modeling which I could deepen in a Ph.D. seminar with Prof. Nick Webber on current theoretical and empir ical fixed-income issues. I also had the opportunity to participate in a Ph.D. seminar with Prof. Yacine Ait-Sahaliah on statistical inference prob lems with discretely observed diffusion models. Finally, I am thankful to Prof. Dr. Herbert Heyer for his introduction to the theory of stochastic par tial differential equations. Last but not least, lowe much more than they imagine to my colleagues at the faculty Dr. Stephan Heilig, Dr. Roman Liesenfeld, Dr. Hartmut Nagel, Dr. Ariane Reiss, and Dr. Jianwei Zhu for the inspiring, fruitful, and pleasant working atmosphere they provided. TUbingen, May 2001 B. Philipp Kellerhals Contents 1 Overview of the Study 1 I Modeling and Estimation Principles 5 2 Stochastic Environment 7 3 State Space Notation 11 4 Filtering Algorithms 15 4.1 Linear Filtering 16 4.2 MMSE ..... . 17 4.3 MMSLE .... . 21 4.4 Filter Recursions 22 4.5 Extended Kalman Filtering 24 5 Parameter Estimation 29 II Pricing Equities 35 6 Introduction 37 6.1 Opening Remarks ....... . 37 6.2 The Case of Closed-End Funds 38 7 Valuation Model 43 7.1 Characteristics of Closed-End Funds 43 7.2 Economic Foundation ...... . 47 7.3 Pricing Closed-End Fund Shares .. . 50 XII Contents 8 First Empirical Results 55 8.1 Sample Data. . . . . 55 8.2 Implemented Model . 60 8.3 State Space Form . . 61 8.4 Closed-End Fund Analysis 64 9 Implications for Investment Strategies 71 9.1 Testing the Forecasting Power .... 71 9.1.1 Setup of Forecasting Study .. 71 9.1.2 Evidence on Forecasting Quality. 73 9.2 Implementing Trading Rules ...... . 75 9.2.1 Experimental Design ...... . 75 9.2.2 Test Results on Trading Strategies 77 10 Summary and Conclusions 83 III Term Structure Modeling 85 11 Introduction 87 11.1 Overview. . . . . . . . . . . . . 87 11.2 Bond Prices and Interest Rates 88 11.3 Modeling an Incomplete Market 92 12 Term Structure Model 97 12.1 Motivation for a Stochastic Risk Premium 97 12.2 Economic Model .............. 100 13 Initial Characteristic Results 105 13.1 Valuing Discount Bonds . . . . . . . . . . . . . . 105 13.2 Term Structures of Interest Rates and Volatilities 112 13.2.1 Spot and Forward Rate Curves 112 13.2.2 Term Structure of Volatilities . . . . . . . 113 13.3 Analysis of Limiting Cases . . . . . . . . . . . . . 116 13.3.1 Reducing to an Ornstein-Uhlenbeck Process 116 13.3.2 Examining the Asymptotic Behavior 118 13.4 Possible Shapes of the Term Structures . 120 13.4.1 Influences of the State Variables. . . 121