Equity derivatives markets [Elektronische Ressource] / von Kai Detlefsen

Detlefsen Kai

Informations

• Publié par: humboldt-universitat_zu_berlin
• Publié le: 01 janvier 2007
• Nombre de lectures: 23
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Equity Derivatives Markets
DISSERTATION
zur Erlangung des akademischen Grades
doctor rerum politicarum
(Doktor der Wirtschaftswissenschaft)
eingereicht an der
Wirtschaftswissenschaftlichen Fakultät
der Humboldt-Universität zu Berlin
von
Dipl.-Math. M.Sc.-Stat. Kai Detlefsen
09.01.1976, Kiel
Präsident der Humboldt-Universität zu Berlin:
Prof. Dr. Christoph Markschies
Dekan der Wirtschaftswissenschaftlichen Fakultät:
Prof. Oliver Günther, PhD
Gutachter:
1. Prof. Dr. Wolfgang Härdle
2. Prof. Dr. Rama Cont
Tag des Kolloquiums: 15. Oktober 2007Abstract
Since the ideas of arbitrage free pricing were born, finance has changed rad-
ically - both in theory and practice. Derivatives markets have evolved and
options serve nowadays as underlyings and as hedging instruments. In this
thesis, we consider some markets for equity derivatives. We start by statisti-
cal analysis of the markets for European options and variance swaps because
theseproductsareimportantforhedgingmorecomplexclaims. Thenwecon-
sider different option pricing models and their calibration to observed price
surfaces. Finally, we investigate the connection between option prices and
the fundamental economic concept of risk aversion by the empirical pricing
kernel.
Keywords:
equity derivatives, implied volatility surface, variance swap, empirical
pricing kernelZusammenfassung
Seit der Entdeckung der arbitragefreien Bewertung hat sich das Gebiet fi-
nance grundlegend geändert - sowohl in der Theorie als auch in der An-
wendung. Märkte für Derivate haben sich entwickelt und Optionen dienen
heutzutage als Basis- und als Absicherungsinstrumente. In dieser Dissertati-
on betrachten wir einige Märkte für Aktienderivate. Wir beginnen mit sta-
tistischen Analysen des Marktes für europäische Optionen und des Marktes
für Varianzswaps, weil diese Produkte die hauptsächlichen Absicherungsin-
strumente für komplexe Optionen sind. Dann betrachten wir verschiedene
Optionspreismodelle und ihre Kalibrierung an beobachtete Preisoberflächen.
SchließlichuntersuchenwirdieVerbindungzwischenOptionspreisenunddem
grundlegenden ökonomischen Konzept der Risikoaversion anhand des empi-
rischen Preiskernes.
Schlagwörter:
Aktienderivate, Implizierte Volatilitätsoberfläche, Varianzswap, PreiskernAcknowledgement
Without the commitment and the energy of my advisor Prof. Dr. Wolfgang
Härdle, this work would not exist in its present form. I would like to ex-
press my deep gratitude to him for constant support and numberless helpful
suggestions.
Moreover, it is a great pleasure for me to thank Prof. Dr. Rama Cont for
two productive and marvellous months at Ecole Polytechnique in Palaiseau
where I found out about interesting new ideas in quantitative finance.
Also I am grateful for the support of Bankhaus Sal. Oppenheim. I would
like to thank Dr. Peter Schwendner and the equity research group for the
enjoyable internships that have inspired my research.
I gratefully acknowledge the financial support of Deutsche Forschungsge-
meinschaft by Sonderforschungsbereich 649 ¨Okonomisches Risiko. Its Fi-
nancial and Economic Data Center made this work possible because of the
unique data and technical support.
I would also like to thank all members of the Institute for Statistics of
Humboldt University Berlin for friendship and encouragement. Finally but
certainly not least I would like to thank my parents, my sister and Valeria
Binello for having been there whenever I needed them.
Berlin im Februar 2007 Kai Detlefsen
iv“Truth is much too complicated to allow anything but approxi-
mations.”
John von Neumann
vIn memory of my mother.
In love, Kai.
viContents
1 Introduction 1
1.1 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . 1
1.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Equity Derivatives Markets 6
2.1 Derivatives Markets . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Underlying assets . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Market for European Options . . . . . . . . . . . . . . . . . . 14
2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Implied volatilities . . . . . . . . . . . . . . . . . . . . 16
2.3.3 Principal components . . . . . . . . . . . . . . . . . . . 19
2.3.4 Cluster analysis . . . . . . . . . . . . . . . . . . . . . . 22
2.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Market for Variance Swaps . . . . . . . . . . . . . . . . . . . . 31
2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.2 Modeling the Term Structure . . . . . . . . . . . . . . 32
2.4.3 Forecasting the Term . . . . . . . . . . . . . 37
2.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Option Pricing Models 53
3.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.1 Local Volatility Models . . . . . . . . . . . . . . . . . . 54
3.1.2 Stochastic Volatility Models . . . . . . . . . . . . . . . 56
3.1.3 Lévy Models . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.4 Market Models . . . . . . . . . . . . . . . . . . . . . . 65
3.1.5 A Semiparametric Stochastic Volatility Model . . . . . 66
3.2 Option Valuation Techniques . . . . . . . . . . . . . . . . . . . 71
3.2.1 Fourier Transforms . . . . . . . . . . . . . . . . . . . . 72
3.2.2 Monte Carlo Simulations . . . . . . . . . . . . . . . . . 74
3.2.3 Partial Differential Equations . . . . . . . . . . . . . . 75
vii4 Estimation 78
4.1 from stock prices . . . . . . . . . . . . . . . . . . . 78
4.1.1 Kalman filter . . . . . . . . . . . . . . . . . . . . . . . 78
4.1.2 Extended Kalman filter . . . . . . . . . . . . . . . . . . 79
4.2 Calibration to option prices . . . . . . . . . . . . . . . . . . . 80
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.2 Models and Data . . . . . . . . . . . . . . . . . . . . . 81
4.2.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2.4 Exotic Options . . . . . . . . . . . . . . . . . . . . . . 90
4.2.5 Model risk . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 104
5 Empirical Pricing Kernels and Investor Preferences 105
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2 Pricing kernels and utility functions . . . . . . . . . . . . . . . 107
5.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3.1 Estimation approaches for the pricing kernel . . . . . . 111
5.3.2tion of the risk neutral density . . . . . . . . . . 113
5.3.3 Estimation of the historical density . . . . . . . . . . . 116
5.3.4 Empirical pricing kernels . . . . . . . . . . . . . . . . . 120
5.4 Individual investors and their utility functions . . . . . . . . . 126
5.4.1 Individual Utility Function . . . . . . . . . . . . . . . . 128
5.4.2 Market Aggregation Mechanism . . . . . . . . . . . . . 130
5.4.3 Estimation of the Distribution of Switching Points . . . 131
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
viiiList of Figures
2.1 Size of over-the-counter and exchange-traded derivatives mar-
kets. (in trillions of US dollar) . . . . . . . . . . . . . . . . . . 7
2.2 Gross size of over-the-counter derivatives markets. (in trillions
of US dollar) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 AutocorrelationofdailyDAXreturns(red,dotted)andsquared
returns (blue, solid), 01/2000 - 06/2004. . . . . . . . . . . . . 10
2.4 Daily DAX returns, - 06/2004. . . . . . . . . . . . . . 11
2.5 Leverage effect in DAX returns (01/2000 - 06/2004) measured
by correlation between lagged returns and squared returns. . . 11
2.6 1 day leverage effect in DAX returns, 01/2000 - 06/2004. . . . 12
2.7 Kernel densities of observed returns (blue, dotted) and sam-
ples of a normal distribution with the same mean and variance
(green, solid). . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.8 Logarithm of kernel densities of observed returns (blue, dot-
ted) and samples of a normal distribution with the same mean
and variance (green, solid). . . . . . . . . . . . . . . . . . . . . 13
2.9 DAX and implied volatility at the money with 1 year to ma-
turity, March 2003 - June 2004. . . . . . . . . . . . . . . . . . 17
2.10 Moneyness/maturity points of option prices on 01 June 2003
(only for moneyness between 0.5 and 1.5). . . . . . . . . . . . 18
2.11 Average implied volatility surface (left) and daily standard
deviation/mean of implied volatility surfaces, Mar 2003 - Jun
2004. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.12 Relative proportion of variance explained by principal compo-
nents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.13 Principal components of daily log implied volatility variation. 23
2.14 Dendrogram of the clusters. . . . . . . . . . . . . . . . . . . . 24
2.15 Meansurfacesofeachcluster(up,left: cluster1; up,right: clus-
ter 2; down,left: cluster 3; down,right: 4). . . . . . . . 25
ix2.16 Principal components of daily log DAX implied volatility vari-
ation, March 200