A Square Root Process for Modelling Correlation

Dissertation

zur Erlangung des akademischen Grades eines Doktor der Naturwissenschaften (Dr. rer. nat.) dem Fachbereich C - Mathematik und Naturwissenschaften - der Bergischen Universit¨atWuppertal vorgelegt von

Cathrin van Emmerich

Promotionsausschuß: Gutachter/Pr¨ufer: Prof. Dr. Michael G¨unther Gutachter/Pr¨ufer: Prof. Dr. Michael Nelles Pr¨ufer: Prof. Dr. Reinhard Michel Pr¨ufer: Prof. Dr. Wil Schilders Diese Dissertation kann wie folgt zitiert werden: urn:nbn:de:hbz:468-20090117 [http://nbn-resolving.de/urn/resolver.pl?urn=urn%3Anbn%3Ade%3Ahbz%3A468-20090117] Acknowledgment

When I started my PhD in 2005, I wanted to learn more about mathematics and finance, gaining deeper knowledge and a better understanding. Now two years later I have more questions than ever before... But I am tremendously grateful that I had the possibility of freedom and shelter at uni- versity to learn and teach, to develop my own answers and ask my own questions. I would like to express my gratefulness at this point knowing that I cannot mention everyone who should be mentioned.

Prof. Dr. Michael G¨unther gave me this opportunity and from the very beginning credit. Thank you very much for this. Also I would like to thank all recent and former members of the Numerical Analysis work- ing group for support, cooperation and company. Particularly I would like to mention Dr. Andreas Bartel. I could always rely on his interest, accuracy, and skills. I had the pleasure to supervise some bright and motivated students at university. Thanks for making my work so enjoyable! Besides I would like to express thank to Prof. Dr. Michael Nelles and Dr. Martin Uzik for letting me profit from their knowledge. Thanks also goes to Prof. Dr. Wil Schilders, who gave me the possibility to spend some months at the TU Eindhoven. Further I would like to thank Prof. Dr. Reinhard Michel, who awoke with his excellent lecture my interest in stochastics. During my PhD I could spend a great time with the equity quant team of Bear Stearns in London headed by Oli Jonsson. It was very valuable to me. Thanks to the whole team, especially to Valer Zetocha. Christian Kahl influenced my PhD years and the thesis itself very strongly and always to the best. Thank you so much. Lastly I would like to thank my parents and Eike for being as they are and doing what they have done. I am very proud of you.

i Contents

Acknowledgment i

Contents ii

1 Introduction 1 1.1 Motivation ...... 1 1.2 Linear correlation coefficient ...... 4 1.3 Outline ...... 6

2 Model 8 2.1 Observed characteristics of correlation ...... 8 2.2 Stochastically correlated Brownian motions ...... 10 2.3 Bounded mean reversion model ...... 12

3 Analytical Properties 14 3.1 Boundary behaviour ...... 14 3.2 Stationary density ...... 20 3.3 Moments ...... 26 3.4 Summary ...... 32

4 Maximum-Likelihood estimator 33 4.1 Overview ...... 33 4.2 Estimating the integral of an Ornstein-Uhlenbeck process ...... 36 4.3 Fitting the stochastic correlation process ...... 40 4.4 Numerical tests ...... 42 4.5 Summary ...... 47

5 Application 48 5.1 Foreign exchange model ...... 48 5.2 Large homogeneous portfolio model (LHPM) ...... 54 5.3 Summary ...... 61

6 Conclusion 62

A Appendix 65

Bibliography 72

Index 75

ii List of Figures

2.1 Correlation between DJI and EURUSD, 1998 - 2005 ...... 9 2.2 Correlation between DJI and its volatility, 02 - 07 ...... 9 2.3 Average correlation in SX5E, 02 - 07 ...... 10 3.1 Restriction on κ for non-attractiveness ...... 20 3.2 Stationary density for varying κ and θ ...... 24 3.3 First three centralised moments of correlation ...... 28 3.4 Recursive computation of moments of integrated correlation ...... 30 3.5 First three centralised moments of realised correlation ...... 31 4.1 Density of integrated correlation process vs. normal density ...... 34 4.2 Density of integrated co