Estimation of continuous-time financial models using high-frequency data [Elektronische Ressource] / vorgelegt von Christian Pigorsch

Christian Pigorsch

Sujets

• Mathematics

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2007
• Nombre de lectures: 12
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

Estimation of Continuous–Time
Financial Models Using
High–Frequency Data
Dissertation an der Fakulta¨t fu¨r Mathematik, Informatik und Statistik der
Ludwig-Maximilians-Universit¨at Mu¨nchen
vorgelegt von
Christian Pigorsch am 1. Februar 2007Ludwig-Maximilians-Universita¨t Mu¨nchen
Fakulta¨t fu¨r Mathematik, Informatik und Statistik
Dissertation
Estimation of Continuous–Time
Financial Models Using
High–Frequency Data
vorgelegt von
Christian Pigorsch
Mu¨nchen, den 1. Februar 2007
Erstgutachter: Prof. Stefan Mittnik, Ph.D.
Zweitgutachter: Prof. Dr. Ludwig Fahrmeir
Externer Gutachter: Prof. A. Ronald Gallant, Ph.D.
Rigorosum: 5. Juni 2007Contents
1 Introduction 8
2 High–Frequency Information 11
2.1 Definition of Realized Variation and Covariation Measures . . . . . 12
2.1.1 Realized Variation . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2 Realized Covariation . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Stylized Facts of Returns and Realized Variation Measures . . . . . 17
2.2.1 Univariate Dataset . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.2 Multivariate Dataset . . . . . . . . . . . . . . . . . . . . . . 24
3 StatisticalAssessmentofUnivariateContinuous–TimeStochasticVolatil-
ity Models 36
3.1 Model Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Affine Models . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 Logarithmic Models. . . . . . . . . . . . . . . . . . . . . . . 39
3.1.3 Jump–Diffusion Models . . . . . . . . . . . . . . . . . . . . . 39
3.1.4 Model Definitions . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Estimation Methodology . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 The General Scientific Modeling Method . . . . . . . . . . . 42
3.3 The Auxiliary Model . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3.1 ADiscrete–TimeModelforDailyReturnsandRealizedVari-
ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.2 Equation–by–Equation Estimation . . . . . . . . . . . . . . 50
3.3.3 System Estimation . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.4 Further Accuracy Checks via Simulations . . . . . . . . . . . 65
3.4 Prior Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4 AMultivariateExtensionoftheOrnstein–UhlenbeckStochasticVolatil-
ity Model 86
4.1 The Univariate Non–Gaussian OU–Type Stochastic Volatility Model 88
4.2 Positive Semidefinite Processes of OU–Type . . . . . . . . . . . . . 90
4.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2.2 Definition and Probabilistic Properties . . . . . . . . . . . . 92
4.2.3 The Integrated Process . . . . . . . . . . . . . . . . . . . . . 95
3Contents
4.2.4 Marginal Dynamics . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 The Multivariate OU–Type Stochastic Volatility Model . . . . . . . 97
4.3.1 Second Order Structure . . . . . . . . . . . . . . . . . . . . 98
4.3.2 State Space Representation . . . . . . . . . . . . . . . . . . 103
4.3.3 Realized Quadratic Variation . . . . . . . . . . . . . . . . . 106
4.4 Estimation Methods and Finite Sample Properties . . . . . . . . . . 108
4.4.1 Estimation Methods . . . . . . . . . . . . . . . . . . . . . . 108
4.4.2 Monte–Carlo Analysis . . . . . . . . . . . . . . . . . . . . . 110
4.5 Empirical Application . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5 Conclusion 125
4List of Tables
2.1 Descriptive Statistics of the Univariate Dataset . . . . . . . . . . . 21
2.2 Company Descriptions of the Multivariate Dataset . . . . . . . . . . 26
2.3 Description of the Multivariate Dataset . . . . . . . . . . . . . . . . 27
2.4 Descriptive Statistics of the Multivariate Dataset (C, INTC, MSFT,
PFE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1 Jump–Diffusion Model Specifications . . . . . . . . . . . . . . . . . 41
3.2 Single–Equation Estimation Results of the Auxiliary Model . . . . . 51
3.3 System Estimation Results of the Auxiliary Model . . . . . . . . . . 64
3.4 Restricted System Estimation Results of the Auxiliary Model. . . . 66
3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6 Estimation Results of the Continuous–Time Stochastic Volatility
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.7 Summary Statistics of Model–Implied Distributions . . . . . . . . . 79
3.8 Summary Statistics of Model–Implied Conditional Distributions . . 82
4.1 Monte–Carlo Results . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2 Univariate Estimation Results for MSFT . . . . . . . . . . . . . . . 116
4.3 Univariate Estimation Results for INTC . . . . . . . . . . . . . . . 117
4.4 Bivariate Estimation Results for MSFT and INTC . . . . . . . . . . 120
4.5 Bivariate Estimation Results for MSFT and INTC, Characteristics . 121
5List of Figures
2.1 Volatility–Signature Plot of the S&P500 Index Futures . . . . . . . 19
2.2 TimeSeriesofReturns, LogarithmicRealizedVariance, Logarithmic
Bipower Variation and Jumps . . . . . . . . . . . . . . . . . . . . . 20
2.3 Unconditional Distributions of Standardized Returns, Logarithmic
Realized Variance, Logarithmic Bipower Variation and Jumps . . . 22
2.4 Sample Autocorrelations and Partial Autocorrelations of Returns,
Logarithmic Realized Variance, Logarithmic Bipower Variation and
Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 News–Impact Curves for Logarithmic Realized Variance, Logarith-
mic Bipower Variation and Jumps . . . . . . . . . . . . . . . . . . . 25
2.6 U–shaped Intraday Patterns . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Autocovariance Function of the Raw and Adjusted Returns . . . . . 30
2.8 Daily Returns and Logarithmic Realized Variances . . . . . . . . . 32
2.9 Daily Realized Correlations . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Residual Analysis of the (log.) Bipower Variation Equation . . . . . 52
3.2 Residual Analysis of the Jump Equation . . . . . . . . . . . . . . . 53
3.3 Residual Analysis of the Return Equation . . . . . . . . . . . . . . 54
3.4 The Volatility of Bipower Variation . . . . . . . . . . . . . . . . . . 55
3.5 Dependency Analysis of the Residuals between the Return Equation
and Bipower Variation Equation . . . . . . . . . . . . . . . . . . . . 58
3.6 Dependency Analysis of the Residuals between the Return Equation
and Jump Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.7 Dependency Analysis of the Residuals between the Bipower Varia-
tion Equation and the Jump Equation . . . . . . . . . . . . . . . . 60
3.8 CDF Scatter Plot of the Single–Equation Innovations . . . . . . . . 61
3.9 CDF Scatter Plot of the System Innovations . . . . . . . . . . . . . 67
3.10 Simulated Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.11 Sample Quantiles of Returns, Logarithmic Realized Variance, Loga-
rithmic Bipower Variation and Jumps . . . . . . . . . . . . . . . . . 71
3.12 Sample Autocorrelations and Partial Autocorrelations of Returns,
Logarithmic Realized Variance, Logarithmic Bipower Variation and
Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.13 Sample Autocorrelations and Partial Autocorrelations of Realized
Variance and Bipower Variation both in Standard Deviation Form . 73
6List of Figures
3.14 Unconditional Distributions of the Mean of the Returns, Realized
Variance and Bipower Variation . . . . . . . . . . . . . . . . . . . . 77
3.15 Unconditional Distributions of the Mean of the Jump Measure, Cor-
relation and the Ljung–Box Statistics . . . . . . . . . . . . . . . . . 78
4.1 Simulated Univariate Sample Path . . . . . . . . . . . . . . . . . . 89
4.2 Simulated Bivariate Sample Path . . . . . . . . . . . . . . . . . . . 99
4.3 Simulated Bivariate Sample Path, Realized Correlation and Scatter
Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4 Simulated Distributions of the Parameter Estimates . . . . . . . . . 112
4.5 Simulated Distributions of Implied Daily Return Characteristics . . 113
4.6 Model–Implied and Empirical Daily Autocorrelation Functions for
MSFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.7 Model–Implied and Empirical Daily Autocorrelation Functions for
INTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.8 Model–ImpliedandEmpiricalDailyAutocorrelationFunctionsBased
on the Bivariate Estimation Results for MSFT and INTC . . . . . . 122
71 Introduction
Modelingthedynamicsofassetpricesandinparticularfinancialvolatilityiscrucial
forderivativepricing, riskmanagementapplications, andassetallocationdecisions.
With the recent availability of high–frequency, or tick–by–tick transaction, data of
variousfinancialmarketstheresearchinthisareahastakennewavenues. Inpartic-
ular, the new information contained in the high–frequency returns is exploited for
example for the direct modeling of these high–frequency returns, as well as