Plenarvorträge - DPG-Tagungen

Symposium Fat-Tail Distributions - Applications from Physics to Finance Donnerstag
Fachvortrag SYFT 2.3 Do 12:30 H1
The Sierpinski signal: 1/f α noise in a simple 1D automaton
model of pattern formation — •Jens Christian Claussen 1 , Jan
Nagler 2 , and Heinz Georg Schuster 1 — 1 Theoretische Physik, Universität
Kiel — 2 Theoretische Physik, Universität Bremen
Cellular automata have been considered widely as a model for complexity
and self-organized criticality: In classic models (Game of Life,
Bak-Sneppen, BTW, OFC), power law spectra and avalanche distributions
have been observed and compared with neural oscillators, extinction
dynamics, and earthquakes [1]. An even more simplified model, yet
exhibiting complex patterns, is the Sierpinski automaton xi(t + 1) =
(xi+1(t) + xi−1(t)) mod 2. Sierpinski patterns appear quite generically
in 1D reaction-diffusion pattern formation and have been observed in
catalytic reactions and in molluscs (Olivia porphyria).
In [2] we investigate the spectral properties of the time dependence
of total activity X(t) = �
i xi(t) of the Sierpinski automaton with initial
condition of one single seed. The spectrum is derived analytically and
shows a power-law decay with exponent 1.15. The relation to the paradigmatic
Thue-Morse sequence is discussed. While in our model the strict
lattice topology is crucial for the analytic result and seems to prevent
a direct mapping on economic agent networks, cellular (and stochastic)
automata may serve as candidates for modelling of fat-tailed observables
resulting from an underlying complex spatio-temporal dynamics.
[1] P. Bak, How nature works; H. J. Jensen, Self-organized criticality.
[2] J. C. Claussen, J. Nagler, and H. G. Schuster, cond-mat/0308277.
Fachvortrag SYFT 2.4 Do 12:45 H1
Robust estimation of correlation matrices and principal component
analysis for fat-tailed elliptical distributions — •Uwe
Jaekel 1 und Gabriel Frahm 2 — 1 NEC Europe Ltd., C&C Research
Laboratories, Rathausallee 10, 53757 St. Augustin — 2 Research Center
CAESAR, Project Group Financial Engineering, Ludwig-Erhard-Allee 2,
53175 Bonn
Many problems in physics and finance require the estimation of covariance
and correlation matrices. We present a maximum likelihood method
for the estimation of these matrices that, unlike the commonly used
moment estimator, is robust for the large class of elliptically contoured
distributions, thus leading to substantially better estimates in the presence
of fat tails. We try to reduce the complexity of the estimation
problem using Bayes or Akaike information criteria, and compare this to
attempts based on random matrix theory. The methods are applied to
high-dimensional simulated and observed equity market data sets. Finally,
we sketch some applications to the analysis of financial markets and
to practical problems like portfolio risk minimisation.
SYFT 3 Fat-Tail Distributions - Applications from Physics to Finance
Zeit: Montag 16:00–18:00 Raum: Poster D
SYFT 3.1 Mo 16:00 Poster D
Extreme value distributions for processes with Gaussian, Lévy
and truncated Lévy distributed increments: A computer simulation
study — •Thomas Schwiertz and Wolfgang Paul —
Institut für Physik, Johannes Gutenberg-Universität, 55099 Mainz
For many applications it is not so much of interest what the average
behavior of a stochastic process is but what its extremal excursions in
a time interval are. Examples are material failure, portfolio risk and the
pricing of barrier options. For sequences of identically distributed random
numbers there exists an exact classification of limiting distributions for
their extremal values. For sum variables (the time-dependent stock price
for instance) no general solution is known. Specifically it is of interest to
contrast Gaussian distributed increments to Lévy distributed ones and
to analyze the crossover between the two for truncated Lévy distributed
increments known to describe stock price fluctuations on the time scale
of a few minutes.
SYFT 3.2 Mo 16:00 Poster D
Pricing of Fire Insurance Contracts — •Magda Schiegl —
Versicherungskammer Bayern, Abt. 8MS02, Maximilianstr. 53, 80530
Muenchen
For pricing of insurance contracts the distribution of the annual claims
losses has to be calculated. We have given empirical data for number of
claims’ occurrence days distribution as well as single claim size distribution
data.
We find a stochastic model describing the number of claims process.
The single claim sizes in fire insurance possess a clearly fat-tailed distribution
function. We combine these two components to obtain the distribution
of annual claims losses We investigate the impact of those two
stochastic components (claim number and size) on the distribution of
annual claims losses, especially the behaviour of the tail.
SYFT 3.3 Mo 16:00 Poster D
Simple stochastic modeling for financial markets — •Hans-
Georg Matuttis — Department of mechanical engineering and intelligent
systems, The University of Electrocommunications, Chofu Chofugaoka
1-5-1,Tokyo 182-8585, Japan
A simple model using a stochastic differential euqation is proposed
which allows to extrapolate from random-walk type distributions to fat
tail distributions. To transcend the purely qualititative aggrement, an attempt
is made to derive the assumption in the model from contemporary
stock market practices in a quantitative way.