State dependent dependencies [Elektronische Ressource] : a continuous time dynamics for correlations / Christoph Becker

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State dependentdependenciesA continuous time dynamicsfor correlationsDissertationSubmitted in Partial Fulfillment of the Requirementsfor the Degree ofDoktor der Wirtschaftswissenschaften(Dr. rer. pol.)Frankfurt School of Finance & ManagementSubmitted byChristoph Becker2010iiMit dem Wissen wächst der Zweifel.Johann Wolfgang GoetheiiiivIch sehne mich nach Freiheitund springemitten im Sprungmöchte ich den Kopf zur Seite drehenund mich im Spiegel sehendenn ich bin schön1wenn ich frei bin.1Winkler [1979], adaptiert.vviContentsContents viiList of Figures ixList of Tables xVorwort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Literature review 11.1 Concepts and quantitative measures for financial asset dependencies . 21.1.1 What do we mean with asset dependencies? What is a crisis? . 21.1.2 Quantitative measures for asset dependencies . . . . . . . . . 31.2 Approaches to model asset dependencies . . . . . . . . . . . . . . . . 51.2.1 Static distributions and the discussion on conditional correla-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 GARCH models . . . . . . . . . . . . . . . . . . . . . . . . 71.2.3 Continuous diffusion and jump diffusion processes . . . . . . 101.2.4 Models with regime-shifts . . . . . . . . . . . . . . . . . . . 121.2.5 Copulas . . . . . . . . . . . . . . . . . . . . . . .

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State dependent
dependencies
A continuous time dynamics
for correlations
Dissertation
Submitted in Partial Fulfillment of the Requirements
for the Degree of
Doktor der Wirtschaftswissenschaften
(Dr. rer. pol.)
Frankfurt School of Finance & Management
Submitted by
Christoph Becker
2010iiMit dem Wissen wächst der Zweifel.
Johann Wolfgang Goethe
iiiivIch sehne mich nach Freiheit
und springe
mitten im Sprung
möchte ich den Kopf zur Seite drehen
und mich im Spiegel sehen
denn ich bin schön
1wenn ich frei bin.
1Winkler [1979], adaptiert.
vviContents
Contents vii
List of Figures ix
List of Tables x
Vorwort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1 Literature review 1
1.1 Concepts and quantitative measures for financial asset dependencies . 2
1.1.1 What do we mean with asset dependencies? What is a crisis? . 2
1.1.2 Quantitative measures for asset dependencies . . . . . . . . . 3
1.2 Approaches to model asset dependencies . . . . . . . . . . . . . . . . 5
1.2.1 Static distributions and the discussion on conditional correla-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 GARCH models . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Continuous diffusion and jump diffusion processes . . . . . . 10
1.2.4 Models with regime-shifts . . . . . . . . . . . . . . . . . . . 12
1.2.5 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.6 Comparison of selected approaches . . . . . . . . . . . . . . 14
2 A continuous time dynamics for correlations 15
2.1 Departing from existing approaches to model asset correlations . . . . 15
2.2 An asset price model with state-dependent correlation in continuous
time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Correlation analyses for discrete time approximations . . . . . . . . . 22
2.4 Empirical analysis of the continuous time model . . . . . . . . . . . . 31
2.4.1 Estimation method . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.2 results . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 How reliable are option prices and hedges under misspecified models? 40
2.6 How is the portfolio variance under a model? . . 48
2.7 International diversification and contagion of financial markets . . . . 52
2.8 Critical review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3 Consistent Pricing of Swaptions and Caps in the Libor Market Model 55
3.1 Departing from existing Libor Market Models . . . . . . . . . . . . . 55
viiContents
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 A Libor Market Model with stochastic correlation . . . . . . . . . . . 58
3.4 A strong distance measure . . . . . . . . . . . . . . . . . . . . . . . 67
3.5 A short digression: Consistent Libor rates with different tenors . . . . 71
3.6 Empirical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.7 A weak distance measure . . . . . . . . . . . . . . . . . . . . . . . . 75
3.8 Comparison with forward Libor correlations used in practice . . . . . 77
Appendices 79
A Reliability of the Euler-based maximum likelihood erstimator 80
B Stochastic Delay Differential Equations 81
Bibliography 88
viiiListofFigures
2.1 Correlation between Deutsche Bank and Commerzbank. . . . . . . . . . 25
2.2 Robust regression results for Deutsche Bank - . . . . . . 26
2.3 Typical plot of correlations versus market state. . . . . . . . . . . . . . . 27
2.4 Non-parametric estimation of correlations versus market state. . . . . . . 27
2.5 The impact of the market memory on correlations and market state. . . . . 28
2.6 The relation between asset volatility and asset trend. . . . . . . . . . . . . 29
2.7 Correlations versus market trend. . . . . . . . . . . . . . . . . . . . . . . 30
2.8 Schematic overview about the behavior of correlations. . . . . . . . . . . 30
2.9 Dependency structures of correlations on the market state. German data. . 33
2.10y of on the market state. US data. . . . 34
2.11 Dependency structures of correlations on the index drift. US data. . . . . 35
2.12 Estimates of the market memory. German data. . . . . . . . . . . . . . . 36
2.13 of the market. US data. . . . . . . . . . . . . . . . . . 37
2.14 Estimates of the market memory for US financials. . . . . . . . . . . . . 38
2.15 Estimations for alternative market states. . . . . . . . . . . . . . . . . . . 39
2.16 Correlations versus the quantile of observed market states. . . . . . . . . 53
3.1 Plot of correlation that depends on forward Libor rates. . . . . . . . . . . 62
3.2 Area where forward Libor rate dependent correlation is well-defined. . . . 62
A.1 Re-estimation of pre-specified dependencies on the market state. . . . . . 80
ixListofTables
2.1 Regression of correlations versus the market state. . . . . . . . . . . . . . 26
2.2 Akaike and Bayesian information criteria for different market states. . . . 40
3.1 Pathwise distance between forward swap rate and Geometric Brownian
Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2 Pathwise distance for stochastic and constant correlations. . . . . . . . . 74
3.3 P in a Libor Market Model with ten forward Libor rates. . 74
3.4 Pathwise distance in a Libor Market with ten forward Libor rates
and another correlation structure. . . . . . . . . . . . . . . . . . . . . . . 75
3.5 Kullback-Leibler distance in a Libor Market Model with ten forward Li-
bor rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.6 K distance for a standard lognormal distribution. . . . . . 77
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