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The equity volatility smile and default risk [Elektronische Ressource] : an empirical analysis of companies of the German DAX-index / Bernd Walter

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Technische Universitat¨ Munchen¨Lehrstuhl fur¨ Betriebswirtschaftslehre - Finanzmanagement und Kapitalmarkte¨Univ.-Prof. Dr. Chr. KasererThe Equity Volatility Smile and Default Risk—An Empirical Analysis of Companies of the German DAX-IndexDipl.-Ing. Bernd WalterVollstandiger¨ Abdruck der von der Fakultat¨ fur¨ Wirtschaftswissenschaften derTechnischen Universitat¨ Munchen¨ zur Erlangung des akademischen Grades einesDoktors der Wirtschaftswissenschaften (Dr. rer. pol.)genehmigten DissertationVorsitzender:Univ.-Prof. Dr. G. FriedlPrufer¨ der Dissertation:1. Univ.-Prof. Dr. Chr. Kaserer2. Univ.-Prof. Dr. R. K. von Weizsack¨ erDie Dissertation wurde am 03.07.2007 bei der Technischen Universitat¨ Munchen¨ ein-gereicht und durch die Fakultat¨ fur¨ Wirtschaftswissenschaften am 07.05.2008 angenom-men.AcknowledgmentsFirst and foremost I would like to thank my dear wife for all her patience and supportthroughout the last year and beyond, especially in times I was too absorbed to fullyassume my role as father and husband.I am also very thankful for the backing of my parents who among other things con-tributed to the continuity of my family income during the last few months.

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Technische Universitat¨ Munchen¨
Lehrstuhl fur¨ Betriebswirtschaftslehre - Finanzmanagement und Kapitalmarkte¨
Univ.-Prof. Dr. Chr. Kaserer
The Equity Volatility Smile and Default Risk

An Empirical Analysis of Companies of the German DAX-Index
Dipl.-Ing. Bernd Walter
Vollstandiger¨ Abdruck der von der Fakultat¨ fur¨ Wirtschaftswissenschaften der
Technischen Universitat¨ Munchen¨ zur Erlangung des akademischen Grades eines
Doktors der Wirtschaftswissenschaften (Dr. rer. pol.)
genehmigten Dissertation
Vorsitzender:
Univ.-Prof. Dr. G. Friedl
Prufer¨ der Dissertation:
1. Univ.-Prof. Dr. Chr. Kaserer
2. Univ.-Prof. Dr. R. K. von Weizsack¨ er
Die Dissertation wurde am 03.07.2007 bei der Technischen Universitat¨ Munchen¨ ein-
gereicht und durch die Fakultat¨ fur¨ Wirtschaftswissenschaften am 07.05.2008 angenom-
men.Acknowledgments
First and foremost I would like to thank my dear wife for all her patience and support
throughout the last year and beyond, especially in times I was too absorbed to fully
assume my role as father and husband.
I am also very thankful for the backing of my parents who among other things con-
tributed to the continuity of my family income during the last few months.
Further, I would like to thank my employer, Siemens Financial Services, and espe-
cially my boss, Peter Rathgeb, also for contributing to the continuity of my family
income and particularly for granting me the flexibility of a sabbatical agreement in
order to accomplish this dissertation.
Very importantly, I wish to thank my professor, Christoph Kaserer, for accepting and
welcoming me as an external doctoral candidate at his chair, for his pragmatic and
uncomplicated mentoring style and finally for the possibility to work in an undisturbed
manner in the rooms of his chair which would not have been possible for me at home.
But my thanks also apply to the chair’s staff and my fellow doctorands for their organi-
zational and technical support as well as their companionship during the last months.
Moreover, I feel obliged to thank Boris Leblanc from the Global Equity and Deriva-
tives Quantitative R&D Team at BNP Paribas for a very fruitful discussion and his
view as expert and practitioner.
Finally I would like to thank Alexander Thomas for his very valuable LaTeX hints
and last but not least my deputy at Siemens Financial Services, Thomas Meurer, for
managing my team during the time of my absence.Contents
Contents I
List of Figures XI
List of Tables XIII
Symbols and Abbreviations XXI
1 Introduction 1
1.1 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Structure of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Summary of the Main Results of the Analysis . . . . . . . . . . . . . 6
2 Financial Valuation Theory 9
2.1 Risk-Neutral Valuation . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 The Concept of Risk-Neutral Valuation . . . . . . . . . . . . 9
2.1.2 Risk-Neutral Valuation and Market Completeness . . . . . . . 10
2.2 The Role of Risk Aversion as a Link between Physical and Risk-
Neutral Probability Density . . . . . . . . . . . . . . . . . . . . . . . 12
3 Option Prices, Volatility Smile, and Risk-Neutral Density 15
3.1 The Volatility Smile Phenomenon . . . . . . . . . . . . . . . . . . . 15
3.1.1 The Black-Scholes Model . . . . . . . . . . . . . . . . . . . 15
3.1.2 The Shortcomings of the Black-Scholes Model and the Volatil-
ity Smile Phenomenon . . . . . . . . . . . . . . . . . . . . . 17
3.1.3 The Relationship between Option Prices, Volatility Smile and
Risk-Neutral Density . . . . . . . . . . . . . . . . . . . . . . 20
3.1.3.1 The Discounted Risk-Neutral Density as the Second
Derivative of a European Call . . . . . . . . . . . . 20
3.1.3.2 The Relationship between the Shape of the Smile
and the Form of the Risk-Neutral Density . . . . . . 22
III Contents
3.1.4 The Role of Risk Aversion in an Idealized Black-Scholes Econ-
omy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Estimation of the Volatility Smile and the Risk-Neutral Density . . . . 24
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.2 Non-Parametric Techniques . . . . . . . . . . . . . . . . . . 25
3.2.2.1 Polynomial Smoothing of the Volatility Smile . . . . 25
3.2.2.2 Kernel Regression . . . . . . . . . . . . . . . . . . 26
3.2.2.3 The Estimation of the Moments of the Risk-Neutral
Density with the Bakshi-Madan Method . . . . . . . 27
3.2.3 Parametric Techniques . . . . . . . . . . . . . . . . . . . . . 29
3.2.3.1 Mixture of Lognormal Distributions . . . . . . . . . 29
3.2.3.2 Expansion Techniques . . . . . . . . . . . . . . . . 30
3.2.4 Other Non-Parametric and Parametric Methods . . . . . . . . 31
3.2.5 Summary and Appraisal of the Available Methods . . . . . . . 32
4 Explanations of the Equity Volatility Smile Excluding Default Risk 35
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Phenomenological Explanations of the Equity Volatility Smile . . . . 36
4.2.1 Stylized Facts of Stock Returns . . . . . . . . . . . . . . . . . 36
4.2.2 Deterministically Changing Volatility in the Stock Price Process 37
4.2.2.1 The CEV Model . . . . . . . . . . . . . . . . . . . 37
4.2.2.2 The Concept of Local Volatility . . . . . . . . . . . 38
4.2.3 Stochastically Changing Volatility in the Stock Price Process . 40
4.2.3.1 A General Representation of a Stochastic Volatility
Model . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.3.2 The Ability of Stochastic Volatility Models to Repli-
cate the Equity Volatility Smile . . . . . . . . . . . 43
4.2.4 Allowing for Jumps in the Stock Price Process . . . . . . . . . 43
4.2.4.1 Jump-Diffusion Processes . . . . . . . . . . . . . . 44
4.2.4.2 Infinite Activity . . . . . . . . . . . . . . 45
4.2.4.3 The Ability of Models Incorporating Jumps to Repli-
cate the Equity Volatility Smile . . . . . . . . . . . 47
4.2.5 Stochastic Interest Rates . . . . . . . . . . . . . . . . . . . . 48
4.3 Causal Explanations of the Equity Volatility Smile . . . . . . . . . . . 49
4.3.1 Market Microstructure and Efficiency Related Explanations . . 50
4.3.2 The Role of Risk Aversion . . . . . . . . . . . . . . . . . . . 52
4.3.2.1 The Manifestation of Risk Aversion in the Real World 52Contents III
4.3.2.2 Fundamental Explanations of the Real World Mani-
festation of Risk Aversion . . . . . . . . . . . . . . 54
4.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 56
5 Default Risk as an Explanation of the Equity Volatility Smile 59
5.1 The Nature of Default Risk and its Theoretical and Empirical Connec-
tion with the Equity Volatility Smile . . . . . . . . . . . . . . . . . . 59
5.1.1 Default Risk in Theory . . . . . . . . . . . . . . . . . . . . . 60
5.1.1.1 Structural Credit Risk Models . . . . . . . . . . . . 60
5.1.1.2 Reduced-Form Credit Risk Models . . . . . . . . . 62
5.1.1.3 The Leverage Effect . . . . . . . . . . . . . . . . . 64
5.1.2 Theoretical Connection between Default Risk and the Implied
Volatility Smile . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1.2.1 Theoretical Connection between Default Risk and
the Level of Equity Volatility . . . . . . . . . . . . 65
5.1.2.2 Theoretical Connection between Default Risk and
the Shape of the Smile . . . . . . . . . . . . . . . . 67
5.1.3 Existing Empirical Evidence of the Relationship between De-
fault Risk and the Equity Volatility Smile . . . . . . . . . . . 68
5.1.3.1 The Leverage Effect . . . . . . . . . . . . . . . . . 68
5.1.3.2 Default Risk and the Level of Equity Volatility . . . 70
5.1.3.3 Default Risk and the Shape of the Equity Volatility
Smile . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.4 Review of the Smile Explanations which are not Related to
Default Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 74
5.1.4.2 Continuous Process Models in the Light of Default
Risk . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.4.3 Discontinuous Process Models in the Light of De-
fault Risk . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Enhanced Credit Risk Models Incorporating the Implied Volatility Smile 77
5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2.2 Structural Credit Risk Models Incorporating the Implied Volatil-
ity Smile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.2.1 Options on a European Call on the Firm’s Assets . . 79
5.2.2.2 on a Down-And-Out Barrier Call on the Firm’s
Assets . . . . . . . . . . . . . . . . . . . . . . . . . 80IV Contents
5.2.2.2.1 Toft/Prucyk Model . . . . . . . . . . . . . 81
5.2.2.2.2 EBIT-Based Model . . . . . . . . . . . . . 82
5.2.2.2.3 Extension of the CreditGrades Model . . . 83
5.2.2.3 Option-Implied Density Approach . . . . . . . . . . 84
5.2.3 Reduced Form Credit Risk Models Incorporating the Implied
Volatility Smile . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2.3.1 Introduction and Overview . . . . . . . . . . . . . . 85
5.2.3.2 Model Mechanics . . . . . . . . . . . . . . . . . . 87
5.2.3.2.1 Models with a Deterministic Stock Price
Diffusion Volatility . . . . . . . . . . . . . 88
5.2.3.2.2 Models with a Stochastic Stock Price Dif-
fusion Volatility . . . . . . . . . . . . . . 90
5.3 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 91
6 Empirical Analysis of the Relationship between the Equity Volatil-
ity Smile and Default Risk 95
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.3 Statistical Methods Used . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3.2 Ordinary Least Squares Regression . . . . . . . . . . . . . . . 98
6.3.2.1 General Assumptions and Properties of the Ordinary
Least Squares Regression . . . . . . . . . . . . . . 98
6.3.2.2 Application of the Ordinary Least Squares Regres-
sion in the Time-Series Domain . . . . . . . . . . . 99
6.3.3 Autocorrelation and Heteroscedasticity in the Residuals and
their Treatment . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3.3.1 Causes and Effects of Autocorrelation and Heteroscedas-
ticity . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3.3.2 Detection of Autocorrelation and Heteroscedasticity 101
6.3.3.3 Treatment of and 102
6.3.4 Non-Stationarity . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3.4.1 Non-Stationarity and its Causes . . . . . . . . . . . 103
6.3.4.2 Detection and Treatment of Non-Stationarity . . . . 104
6.3.4.2.1 Detection and Treatment of a Time Trend . 104
6.3.4.2.2 and T of a Stochastic
Trend . . . . . . . . . . . . . . . . . . . . 104Contents V
6.4 Data Processing and Conditioning . . . . . . . . . . . . . . . . . . . 107
6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4.2 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4.3 Computation of Historical Moments . . . . . . . . . . . . . . 111
6.4.3.1 Observation Period and Frequency . . . . . . . . . . 111
6.4.3.2 Generation of the Return Series . . . . . . . . . . . 112
6.4.3.3 Computation of Moments . . . . . . . . . . . . . . 112
6.4.4 Computation of Implied Volatilities . . . . . . . . . . . . . . 113
6.4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 113
6.4.4.2 Input Variables for the Option Evaluation Function . 113
6.4.4.3 Data Quality Requirements . . . . . . . . . . . . . 114
6.4.4.4 Option Price Evaluation with the Cox-Ross-Rubinstein
Binomial Tree . . . . . . . . . . . . . . . . . . . . 115
6.4.4.5 Root-finding Algorithm . . . . . . . . . . . . . . . 115
6.4.5 The Smoothing of the Implied Volatility Smile . . . . . . . . 116
6.4.5.1 Assessment of the Available Implied Volatilities . . 116
6.4.5.2 Quadratic Polynomial as the Method of Choice for
the Smoothing of the Implied Volatility Smile . . . . 119
6.4.5.3 Regression Model . . . . . . . . . . . . . . . . . . 121
6.4.5.4 Re Results . . . . . . . . . . . . . . . . . . 122
6.4.5.4.1 Regression Results and Regression Statistics122
6.4.5.4.2 Interpretation of the Regression Results . . 125
6.4.6 Computation of Risk-Neutral Skewness and Kurtosis . . . . . 127
6.4.6.1 Qualitative Assessment of the Risk-Neutral Return
Density . . . . . . . . . . . . . . . . . . . . . . . . 127
6.4.6.2 Computation of the Moments of the Risk-Neutral
Return Density . . . . . . . . . . . . . . . . . . . . 129
6.4.6.2.1 Practical Issues and Problems Concerning
the Implementation of the Bakshi-Madan
Method . . . . . . . . . . . . . . . . . . . 129
6.4.6.2.2 Modification of the Bakshi-Madan Method
. . . . . . . . . . . . . . . . . . . . . . . 130
6.4.7 The Variables in the Regression Analysis . . . . . . . . . . . 133
6.4.7.1 Generation of Constant Maturity Smile, Skewness
and Kurtosis . . . . . . . . . . . . . . . . . . . . . 133
6.4.7.2 Synopsis of the Variables . . . . . . . . . . . . . . 133
6.4.7.3 Summary Statistics of the Variables . . . . . . . . . 136VI Contents
6.4.7.3.1 Summary Statistics of All Variables . . . . 136
6.4.7.3.2 of Specific Variables
Deserving a Closer Examination . . . . . . 137
6.4.7.4 Assessment of the Stationarity of the Variables . . . 140
6.5 Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.5.1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . 143
6.5.1.2 Robustness Checks . . . . . . . . . . . . . . . . . . 144
6.5.1.2.1 The Issue of Possible Non-Stationarity . . 144
6.5.1.2.2 Robustness in the Presence of Additional
Explanatory Variables . . . . . . . . . . . 144
6.5.1.3 Structure of the Empirical Analysis . . . . . . . . . 146
6.5.2 The Leverage Effect . . . . . . . . . . . . . . . . . . . . . . 146
6.5.2.1 Derivation of the Regression Model . . . . . . . . . 147
6.5.2.2 Regression Results . . . . . . . . . . . . . . . . . . 148
6.5.2.2.1 Examination in the Time Series Dimension 148
6.5.2.2.2 in the Framework of a Pooled
Series Regression . . . . . . . . . . . . . 151
6.5.2.3 Robustness of the Regression Results . . . . . . . . 152
6.5.2.3.1 Robustness Check of the Time Series Re-
gression Results . . . . . . . . . . . . . . 152
6.5.2.3.2 Robustness of the Pooled Series Regres-
sion Results . . . . . . . . . . . . . . . . 156
6.5.2.4 Summary and Discussion of the Results in the Light
of Hitherto Existing Findings . . . . . . . . . . . . 157
6.5.2.4.1 Summary of the Results . . . . . . . . . . 157
6.5.2.4.2 Discussion of the Results in the Light of
Hitherto Existing Findings . . . . . . . . . 157
6.5.3 The Credit Spread Level and the Implied Volatility Level . . . 159
6.5.3.1 Derivation of the Regression Models . . . . . . . . 159
6.5.3.2 Regression Results . . . . . . . . . . . . . . . . . . 159
6.5.3.2.1 Examination in the Time-Series Dimension 159
6.5.3.2.2 in the Framework of a Pooled
Series Regression . . . . . . . . . . . . . 163
6.5.3.3 Robustness of the Regression Results . . . . . . . . 164
6.5.3.3.1 Examination in the Time-Series Dimension 164