GARCH-like models with dynamic crash-probabilities [Elektronische Ressource] / Paul Koether

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• Mathematics

Informations

• Publié par: technische_universitat_kaiserslautern
• Publié le: 01 janvier 2005
• Nombre de lectures: 32
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

GARCH{like Models with Dynamic
Crash-Probabilities
Paul Koether
Vom Fachbereich Mathematik der
Technischen Universit at Kaiserslautern
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr. rer. nat.)
genehmigte Dissertation
Erster Gutachter: Prof. Dr. Jurgen Franke
Zweiter Gutachter : Prof. Dr. Jens{Peter Krei
Datum der Disputation: 30.09.2005
D 386Abstract
We work in the setting of time series of nancial returns. Our starting point
are the GARCH models, which are very common in practice. We introduce the
possibility of having crashes in such GARCH models. A crash will be modeled by
drawing innovations from a distribution with much mass on extremely negative
events, while in "normal" times the innovations will be drawn from a normal dis-
tribution. The probability of a crash is modeled to be time dependent, depending
on the past of the observed time series and/or exogenous variables. The aim is
a splitting of risk into "normal" risk coming mainly from the GARCH dynamic
and extreme event risk coming from the modeled crashes.
We will present several incarnations of this modeling idea and give some basic
properties like the conditional rst and second moments. For the special case
that we just have an ARCH dynamic we can establish geometric ergodicity and,
thus, stationarity and mixing conditions. Also in the ARCH case we formulate
(quasi) maximum likelihood estimators and can derive conditions for consistency
and asymptotic normality of the parameter estimates.
In a special case of genuine GARCH dynamic we are able to establish L -1
approximability and hence laws of large numbers for the processes itself. We
can formulate a conditional maximum likelihood estimator in this case, but can-
not completely establish consistency for them.
On the practical side we look for the outcome of estimating models with genuine
GARCH dynamic and compare the result to classical GARCH models. We ap-
ply the models to Value at Risk estimation and see that in comparison to the
classical models many of ours seem to work better although we chose the crash
distributions quite heuristically.
iAcknowlegement
Many thanks to my supervisor Prof. Jurgen Franke for providing the topic and
much support. Thanks to Dr. Jean Pierre Stockis for critical comments, support
and encouragement. I’m indebted to my brothers in arms Dr. Joseph Tadjuidje
Kamgaing and Dr. Charles Andoh for many mathematical discussions and mu-
tual help. Many thanks to Mrs. Beate Siegler the secretary of our working
group for friendliness and support in non matters. The nancial
and logistic support of the Graduiertenkolleg Mathematik und Praxis is highly
appreciated.
iiContents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Acknowlegement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Introduction 1
Motivation and modeling idea . . . . . . . . . . . . . . . . . . . . . . . 1
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1 GARCH processes 4
1.1 Financial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 ARCH{models . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 GARdels . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 ARMA{GARCH and GARCH in mean . . . . . . . . . . . 6
1.2.4 T{GARCH and E{GARCH . . . . . . . . . . . . . . . . . 6
1.3 GARCH models with Markov{switching . . . . . . . . . . . . . . 7
1.4 Estimation and theoretical properties . . . . . . . . . . . . . . . . 7
1.4.1 GARCH(p,q) . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.2 The Markov switching model . . . . . . . . . . . . . . . . 10
1.5 Methods of the Theory . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Non normalized GARCH . . . . . . . . . . . . . . . . . . . . . . . 12
2 A rst model with probability of a crash 16
2.1 De nition of the model . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Expected Value, Variance, Covariances . . . . . . . . . . . . . . . 17
2.2.1 Expected value . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.2 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3 Covariances . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 The choice of the crash distribution and the problem of de ning a
crash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Higher Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.1 Skewness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.2 Kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
iii2.7 Transformations of the signum{function as Crash{probabilities . . 26
2.8 Models with crash probabilities depending on external variables . 29
2.8.1 Mean and Variance . . . . . . . . . . . . . . . . . . . . . . 29
2.8.2 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.8.3 Discussion of SLLN for mixing processes . . . . . . . . . . 33
3 Modi cations of the model 35
3.1 Crash-models with GARCH-volatilities . . . . . . . . . . . . . . . 35
3.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 An alternative view of the model . . . . . . . . . . . . . . . . . . 39
3.3.1 A di eren t view of the introduced models . . . . . . . . . . 40
3.3.2 A generalization . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 An alternative model . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Asymptotics in the pure ARCH case 44
4.1 Markov Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Some Markov Chain Theory . . . . . . . . . . . . . . . . . . . . . 45
4.3 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4 Stationarity in the pure ARCH case of the original model (the
CARCH-S model) . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 Stationarity in the pure ARCH case of the alternative volamodel
(the ACARCH-V model) . . . . . . . . . . . . . . . . . . . . . . . 51
5 Estimation 57
5.1 The general model . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 The methodology of P otscher and Prucha (1997) . . . . . . . . . . 58
5.2.1 Local law . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3 Application to our context . . . . . . . . . . . . . . . . . . . . . . 61
6 Asymptotic properties of di eren t Quasi Maximum Likelihood
Estimators in the pure ARCH case 64
6.1 Consistency of a Quasi{Maximum Likelihood Estimator in the
CARCH-S model . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2 results for the Conditional Maximum Likelihood esti-
mator of the CARCH-S model . . . . . . . . . . . . . . . . . . . . 71
6.3 Consistency of a Quasi{Maximum Likelihood Estimator in the
ACARCH-V model . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.4 results for the Conditional Maximum Likelihood esti-
mator of the ACARCH-V model . . . . . . . . . . . . . . . . . . . 83
6.5 Some generic results concerning Asymptotic Normality . . . . . . 86
6.6 Asymptotic Normality and alternative Consistency for the Quasi{
Maximum Likelihood Estimator in the ACARCH-V model . . . . 88
iv6.7 Asymptotic Normality and alternative Consistency for the Quasi{
Maximum Likelihood Estimator in the CARCH-S model . . . . . 95
6.8 Normality Results for the Maximum Likelihood Esti-
mator in the CARCH-S Model . . . . . . . . . . . . . . . . . . . . 98
6.9 Asymptotic Normality of the Maximum Likelihood Estimator in
the ACARCH-V model . . . . . . . . . . . . . . . . . . . . . . . . 105
6.10 Chiastic complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7 Some Approximation Properties 110
7.1 Moment Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.2 L -Approximability . . . . . . . . . . . . . . . . . . . . . . . . . . 1111
7.3 Near Epoch Dependence . . . . . . . . . . . . . . . . . . . . . . . 113
8 Some results concerning asymptotics and inference in a restricted
model with real GARCH dynamic 115
8.1 A restricted model and its approximability by mixing processes . 115
8.2 Steps towards asymptotic theory of estimation . . . . . . . . . . . 118
9 Estimation and simulation in the CGARCH-S and the CGARCH-
V model 120
9.1 The models we used and corresponding notation . . . . . . . . . . 120
9.2 Estimation and simulation of models without external variables . 121
9.3 A rst study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
9.4 Some models of CGARCH-S type . . . . . . . . . . . . . . . . . . 125
9.5 The CGARCH-V model . . . . . . . . . . . . . . . . . . . . . . . 131
10 Estimation in models with additional external parameters in the
crash probability 133
10.1 A threshold model . . . . . . . . . . . . . . . . . . . . . . . . . . 133
10.2 Models with additional external variables . . . . . . . . . . . . . . 136
10.3 Using just external variables . . . . . . . . . . . . . . . . . . . . . 139
10.4 Estimation of simulated models . . . . . . . . . . . . . . . . . . . 143
11 Practical results concerning the Val