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Mean Reversion Models of Financial Markets
Inaugural–Dissertation
zur Erlangung der Wurde¨ eines Doctor rerum politicarum
an der
Universit¨ at Bremen
Fachbereich 7 — Wirtschaftswissenschaft
vorgelegt von: Eric Hillebrand
e–mail: erhil@aol.com
Gneisenaustraße 48
28201 Bremen
Telefon: 0421-550531.
Betreuer: Prof. Dr. Otto Steiger
Institut fur¨ Konjunktur- und Strukturforschung (IKSF)
Stanford, 12. Mai 2003Acknowledgments
I am grateful to my adviser, Otto Steiger, for continuous support and advise
during the last six years. His unconditional commitment and confidence in the
success of this project was an invaluable source of motivation.
This thesis was written in Bremen and Stanford. From 1999 to 2001 I was
at the Center for Complex Systems and Visualization (CeVis) in Bremen. It
is a pleasure to thank Heinz-Otto Peitgen, Carl Evertsz and the team of CeVis
and MeVis for creating a first rate working environment. Discussions with Carl
Evertsz, Ralf Hendrych, Sven Kohle, Richard Rascher-Friesenhausen and Peter
Singer were fruitful.
The support of Ludwig Arnold at the Department of Mathematics at the
University of Bremen was crucial in many ways. He gave advise on early stages
of the model for mean reversion in returns and his excellent contacts facilitated
my way over the Atlantic.
I spent the academic years 2001/2002 and 2002/2003 at the Department of
Mathematics at Stanford University. I am deeply indebted to George Papanico-
laou, who invited me, gave intense and very motivating advise and, last but not
least, supported me when I went on the U.S. job market for economists. This
would have been impossible without him.
For many helpful discussions I thank Caio Almeida, Arnold Kim, Mordecai
Kurz and the participants of his Ph.D. student’s seminar at the Stanford Depart-
ment of Economics, as well as Jonathan Mattingly and Knut Sølna.
Any remaining errors are mine, of course.
Kein Unternehmen wie dieses ohne Familie und Freunde: Vielen Dank an
Marita, Edgar, Alexandra, Anni; Gun¨ ter und Irmtraut (wo lasst¨ man sein Zeugs
wenn man fur¨ einige Jahre ins Ausland geht?); Mareike, Thomas, Ling, Christo-
pher, Anita, Arnold, Caio, Ellen, Vero. Katrin, und ohne sie hoc¨ hstwahrscheinlich
alles ganz anders.
3This analysis suggests that a more catholic approach should be taken to explaining
the behavior of speculative prices.
Lawrence H. SummersContents
1 Introduction: Mean Reversion in Stock Market Prices 17
A Mean Reversion in Prices and Returns 27
2 Measuring and Interpreting Mean Reversion in the Data 29
1 DefinitionsofMeanReversion .................... 29
2 EfficientMarketsandMeanReversion ............... 32
3 RationalesofMeanReversion 37
4 MeasurementandEvidenceofMeanReversion........... 40
5 Conclusion. .............................. 50
3 A Mean Reversion Theory of Stock Market Crashes 53
1 MeanReversionandStockMarketCrashes . ............ 53
2 A Mean Reversion Theory of Stock Market Crashes . . . . . . . . 54
a) MeanReversionExpectations ................ 54
b) Mean Reversion Illusions and Disillusions . . . . . . . . . . 56
c) Mean Reversion Disillusion and October 19, 1987 . . . . . 59
3 AMeanReversionModelforStockReturns ............ 61
4 Mean Reversion and the Stock Market Crash of 1987 in Market
Data . ................................. 63
a) The Mean Reversion Disillusion . . . . . . . . . . . . . . . 65
b) The Mean Rev Illusion . . . . . . . . . . . . . . . . . 68
5 A Note on the Stock Market Crash of 1929 . . . . . . . . . . . . . 73
6 Conclusions .............................. 74
B Mean Reversion in Volatility 77
4 Volatility Persistence, Mean Reversion, and Long Memory 79
5 Mean Reversion and Persistence in GARCH(1,1) 85
1 Time Scales and Persistence in Financial Volatility Data . . . . . 85
2 PersistenceEstimationwithGARCHModels............ 87
78 CONTENTS
a) TheModelFormulation . .................. 87
b) Measures of Persistence and Mean Reversion . . . . . . . . 88
c) Maximum Likelihood Estimation . . . . . . . . . . . . . . 89
d) GARCH(1,1) and Market Data: High Persistence in the
Volatility of the Dow Jones and S&P500 . . . . . . . . . . 90
e) HighPersistenceasaStylizedFact ............. 91
3 Parameter Changes and Global GARCH(1,1) Estimations . . . . . 93
a) The Geometry of Almost-Integration . . . . . . . . . . . . 93
b) TheAnalysisofAlmost-Integration . ............ 94
c) Simulations . ......................... 101
4 Estimation of the Short Scale in Stock Volatility . . . . . . . . . . 103
a) SyntheticData . ....................... 106
b) MarketData . ........................ 107
5 SummaryandConclusion . ..................... 108
6 Generalization to GARCH(p,q) 111
1 Multiple Scales and Higher Order ARMA and GARCH . . . . . . 111
2 MeasuringMeanReversioninARMA(1,1) ............. 112
3 AggregationofARMA(1,1)Models ................. 113
4 AggregationofGARCH(1,1)Models . ............... 114
5 Unknown Parameter Regime Changes and Global GARCH(p,q)
Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Simulations .............................. 126
7 Conclusion. 129
7 The Impact of Japanese Foreign Exchange Intervention on Level
and Volatility of the Yen/Dollar Exc Rate 131
1 Sterilized Intervention and Volatility . . . . . . . . . . . . . . . . 131
2 The Discussion of the Japanese Interventions in the Literature . . 132
3 Data . ................................. 135
4 A GARCH(p,q) Model with Interventions as Exogenous Variables 137
5 EstimationResults .......................... 138
6 ChangepointDetection . ....................... 139
7 Conclusion. .............................. 141
8 Conclusions and Directions 143
1 Summaries............................... 144
2 FutureResearch. ........................... 148
Bibliography 151
Appendix I 158CONTENTS 9
Appendix II 160