Table Of ContentLecture Notes in Economics
and Mathematical Systems 506
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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
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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
