Abstract:
The constant conditional correlation models are versatile tools for modelling
the behaviour of financial returns. Mathematically speaking, they are solutions
of a special class of stochastic difference equations (s.d.e.). The extremal
behaviour of general solutions of s.d.e. has been studied in detail in Kesten (1973)
and Perfekt (1997). The spectral measure of the extreme value distribution is
the central concept to understanding the joint extremal behaviour of the multivariate time series. In this talk we propose an estimator for the spectral measure
and prove its consistency. Our estimator is the tail empirical measure associated with the multivariate time series. Successful use of the estimator depends on
a good choice of k, the number of upper order statistics contributing to the
empirical measure. We introduce a new criteria for the choice of k based on a
scaling property of the spectral measure. We investigate the performance of our
estimation technique on time series from HFDF96 data set. We use the estimated spectral measure to calculate probabilities of large movements in a currency exchange rate conditional on the occurrence of extreme returns in another related exchange rate. We are able to prove a high level of correlation in the extremes for most of the currencies in the EU. We also quantify the changes in the level of
correlation in the extreme as the sampling frequency decreases. We are able to
show that, for most of the European currencies, the level of correlation is very
high even at 4-hour sampling frequency.
Abstract:
Several attempts have been made to remedy some of the shortcomings of the
Black-Scholes model by replacing Brownian motion by a process with correlated
increments. In the talk we discuss two Gaussian processes whose increments
exhibit long-range dependence and their applicability to option pricing.
Tuesday, June 23, 1998, 12.15 h (Hermann-Weyl-Zimmer, ETHZ, HG G43)
Abstract:
We will present a notion of randomness for sequences of arbitrary length,
particularly short sequences, called ApEn. As an important property,
ApEn allows us to order sequences by randomness (as for example
heartbeat rates or the release process of hormones).
Since ApEn results only in a partial ordering, we will investigate the
existence of maximally irregular sequences. We will show that maximal
irregularity (randomness) with respect to ApEn coincides with the notion
of equidistributuion (or k-distribution), as known from pseudorandom number generators.
Tuesday, June 23, 1998, 15.15 h (IFW A 32, Informatik, Clausiusstr. 59, 8006 Zürich)
(Seminar über Statistik, ETH and University of Zürich )
Tuesday, June 23, 1998, 17.15 h (ETHZ, HG D 7.1)
Peter Grandits
(Universität Wien)
p-optimal martingale measures and its asymptotic relation with
the minimal entropy measure
Abstract:
Martingale measures, which minimize the L2-norm
(variance-optimal), respectively
the entropy of the density with respect to the original measure,
have been extensively studied in the literature. We give a connection
between these concepts by introducing the so called p-optimal
measures. Indeed we prove that the minimal entropy measure can be
seen as the limit of the p-optimal ones. This is done in a discrete
time setting and then also for continuous processes.
I will present the easy but technical proof of this result, explain the
relation with the passport option and if time permits, give some links to
martingale inequalities.
Thursday, June 4, 1998, 17.15 h (Hermann-Weyl-Zimmer, ETHZ, HG G43)
Abstract:
We extend two known (elementary?) properties of beta and gamma distributions:
the sum of independent gammas is also gamma if the second parameters
are the same (additive property);
when the parameters are well chosen, the product of a beta and a gamma
turns out to be a gamma (multiplicative property).
We find that: for any a,b>0, the distribution of UG(a)+G(b) is the same as
that of G(a+b)[1+(U-1)B(a,b)]. Here the G's are independent gamma distributions
with the same second parameter, and B(a,b) is beta with paramaters a and b, and
U is any variable independent of all the others.
Some consequences are:
two new affine properties of beta and gamma distributions,
limit distributions of Markov chains,
distributions of non-homogeneous Markov chains,
limit distributions of time series with random coefficients,
distributions of some discounted sums.
Wednesday, June 3, 1998, 17.15 h (Auditorium Maximum, ETHZ, HG F30)
Abstract:
This talk will present pricing and risk simulation
models for credit risk that involves the timing of different credit
events, such as defaults or downgrades. A classic
application is valuation of a credit derivative that pays the loss on the
first (or first n) defaults of a collection of obligations. Another
application is the simulation of default losses on a portfolio of OTC
derivatives. Computational methods that are tractable and allow for
correlated default risk are emphasized.
Tuesday, May 12, 1998, 12.15 h (Hermann-Weyl-Zimmer, ETHZ, HG G43)
Alexander McNeil
(ETH Zürich)
Conditional quantiles and value at risk for heavy-tailed financial time series
Abstract:
We suggest a new method for calculating conditional quantiles
of financial time series using a combination of
GARCH modelling and extreme value theory. We use this
method to calculate VaR for a number of real return series.
(Joint work with Rüdiger Frey.)
Monday, May 11, 1998, 13.15 h (Hermann-Weyl-Zimmer, ETHZ, HG G43)
