Abstract:
This paper proposes an approach to the intraday analysis of diversified world
accumulation indices. The growth optimal portfolio (GOP) is used as reference
unit or benchmark in a continuous financial market model. Diversified global
portfolios, covering the world financial market, are constructed and shown to
approximate the GOP. The normalized GOP is modeled as a time transformed square
root process of dimension four. These dynamics are empirically verified in a
robust manner for several world stock indices. Furthermore, the long-term
evolution of the transformed time is modeled via a constant net growth rate of the
drift of the discounted GOP and a quickly evolving market activity. The latter is
decomposed into a mean reverting stochastic market activity process and a
deterministic seasonal market activity component. The empirical findings identify
a simple and realistic model for a world stock index that reflects its historical
evolution reasonably well by using only a few constant parameters.
(Joint work with Leah Kelly and
Eckhard Platen)
(Seminar on Financial and Insurance Mathematics)
Wednesday, July 2, 2003, 17.15 h
(ETHZ,
Auditorium Maximum, HG F30)
Abstract:
We consider the problem of estimating the tail index parameter from an i.i.d.
sample. The aim of the talk is to give a natural resolution to the
'Hill horror plot' paradox and to rehabilitate the Hill estimator, for finite
sample sizes, by looking at the problem from the point of view of selecting an
appropriate tail.
We give a new adaptive method for selecting the number of upper order statistics
used in the estimation of the tail of a distribution function. The selection
procedure consists in consecutive testing for the hypothesis of homogeneity of
the estimated parameter against the change-point alternative. The selected
number of upper order statistics corresponds to the first detected change-point.
Our main results are non-asymptotic and state optimality of the proposed method.
A performance of the method is illustrated by a simulation study. This is joint
work with joint with
Ion Grama
(Vannes, France).
(Seminar on Financial and Insurance Mathematics)
Tuesday, June 10, 2003, 12.15 h
(ETHZ,
Hermann-Weyl-Zimmer, HG G43)
Abstract:
We deal with a version of the duality theorem of convex optimization. As a
special case, we obtain a general version of the duality theorem of the problem
of utility maximization in the context of mathematical finance. In particular,
we are in a position to generalize results by Kramkov and Schachermayer (1999),
Cvitanic, Schachermayer and Wang (2001) and Zitkovic (2002). In contrast to
these results, the utility function under consideration satisfies only very weak
regularity resp. growth conditions. Moreover, we do not assume that the domain
of the problem of utility maximization is closed (in the context of mathematical
finance, this would be equivalent to the existence of an equivalent martingale
measure). In order to prove the main theorem, we use different arguments than
the ones in the above-mentioned papers. However, we take on some important ideas
of the original proofs by Kramkov and Schachermayer (1999).
(Seminar on Financial and Insurance Mathematics) Thursday, June 5, 2003, 17.15 h
(ETHZ,
Hermann-Weyl-Zimmer, HG G43)
Abstract:
Discrete 'Brownian motion', stochastic differential equations with respect to
discrete Brownian motion, and the 'predictable' Itô formula for the solution
will be presented. Applications are (a) numerical calculations,
(b) spectral analysis and (c) insights.
Abstract:
We present the quantization method which is well adapted for the pricing and
hedging of American options on a basket of assets. Its purpose is to compute
a large number of conditional expectations by projection of the diffusion on
an optimal grid designed to minimize the (mean square) projection error. The
basic algorithm employs piecewise constant approximations of the functions
at hand. We also give an improvement of this algorithm which amounts to
construct first order schemes. This means that one employs piecewise linear
approximations. In order to do it we add some correctors computed by a
Malliavin calculus method on one hand, and we use the optimality of the grid
on the other hand. Finally we compute Greeks using the quantization
algorithm in connection with Malliavin calculus.
The talk is based on the preprints
First order schemes in the
numerical quantization method and
A quantization tree method
for pricing and hedging multi-dimensional American options,
both jointly with Gilles Pagès and Jacques Printems.
Abstract:
We describe a general method to model dynamic dependence between components of
multidimensional Lévy processes by introducing Lévy copulas. These
objects have the same properties as copulas but a different domain of definition
(the entire plane rather than the unit square). They can be used to separate
dependence from the behavior of the components of a multidimensional Lévy
process. New multidimensional Lévy process based models can be
constructed by taking arbitrary one-dimensional processes and linking them using
a Lévy copula with desired dependence pattern. We construct
parametric families of Lévy copulas and develop an algorithm for
simulating multidimensional dependent Lévy processes using their
Lévy copulas. Finally we illustrate our method by showing how it can be
used to build multidimensional models with jumps for finance and insurance.
(Seminar on Financial and Insurance Mathematics)
Announcement:Workshop on Financial Time Series,
TU Munich in Garching, May 28, 2003.
Tuesday, May 27, 2003, 12.15 h
(ETHZ,
Hermann-Weyl-Zimmer, HG G43)
Abstract:
The chaotic approach to interest rates introduced recently by Hughston
and Rafailidis is an elegant and concise way to build new models and to
understand old ones. The Cox-Ingersoll-Ross model ought therefore to
have a natural chaotic representation, but until now it has not be
found. After a review of various approaches to interest rate modelling,
I will determine one chaotic representation for the extended CIR model
in "integer dimensions". This does have natural features, provided one
accepts some technology originating in quantum field theory. In the
end, I will introduce a simpler but related class of models which may
prove to be deserving of further study. This is joint work with
Matheus Grasselli.
(Seminar on Financial and Insurance Mathematics)
Monday, May 19, 2003, 19.30 h
(Aula,
Universität-Zentrum,
Rämistr. 71)
Abstract:
We discuss a new approach to pricing some exotic financial options
(of lookback and barrier types). The approach is based on
employing Spitzer's factorization identity for the distribution of
the maximum in a random walk. It can be used when the underlying
asset is modelled by a geometric Lévy process. We demonstrate
that using Spitzer's identity along with Fourier transforms
enables one to compute the prices of the discrete lookback and
barrier options via a simple recursive formula. We also discuss
the problem of pricing of time-dependent barrier options when
interest rate and volatility are given functions in Black-Scholes
framework. The calculation of the fair price reduces to the
calculation of non-linear boundary crossing probabilities for a
standard Brownian motion. The proposed method is based on a
piecewise-linear approximation for the boundary and repeated
integration. The numerical example provided draws attention to the
performance of suggested method in comparison to some alternatives.
(BAPSSeminar)
Thursday, May 8, 2003, 17.15 h
(ETHZ,
Hermann-Weyl-Zimmer, HG G43)
Abstract:
Generalized Ornstein-Uhlenbeck processes are used in engineering
and financial applications (e.g. credit risk, interest rate
models) under different assumptions about the jump component. In
this talk, we discuss several approaches for studying the
distribution of first passage times of Ornstein-Uhlenbeck
processes with a jump component. These approaches include:
integro-differential equations, martingale techniques and
Monte-Carlo simulation employing variance-reduction techniques. In
particular, based on martingale considerations, we derive the
exponential and the normal approximations for the distribution in
question, and compare the accuracy of these approximations with
the results of Monte-Carlo simulation.
(Seminar on Financial and Insurance Mathematics)
Wednesday, May 7, 2003, 17.15 h
(ETHZ,
Auditorium Maximum, HG F30)
Abstract:
Gamma constraint on a portfolio is a restriction on the
variations of the portfolio process with respect to the changes in
the underlying asset. The minimal super-replication cost of any contingent
claim is the smallest initial capital from which, by using a portfolio
process, one can super hedge the claim at maturity, with probability one.
Using dynamic programming techniques, we show that minimal super-replication
cost is the viscosity solution of a quasi-variational inequality, which
involves a constraint for the Hessian of the minimal cost function.
This equation is the parabolic majorant of the incorrect, non-parabolic
equation one guesses for this problem. Crucial step is a detailed asymptotic
analysis of double stochastic integrals.
(This is joint work with
P. Cheridito from ETH
and N. Touzi
from Paris.)
(Seminar on Financial and Insurance Mathematics)
Tuesday, April 29, 2003, 12.15 h
(ETHZ,
Hermann-Weyl-Zimmer, HG G43)
Abstract:
Spectrally negative Lévy processes appear in the theory of insurance risk,
financial mathematics as well as queuing theory. Fluctuation theory of
this class of processes has been well developed over the last 30 or so
years by a number of authors. In this mini-lecture course we shall give
new, straightforward proofs of some fundamental fluctuation identities
concerning exit problems of spectrally negative Lévy processes. The proofs
rely for the most part on (elementary properties of) martingales and the
strong Markov property. The programme should therefore be accessible to
non-specialists.
(An accompanying 11-page document
to the lectures is available online.)
(Mathematisches Kolloquium Zürich)
Announcement:Workshop I/2003
of CARISMA on Friday, April 11, 2003,
from 10.00 to 17.00 in
Credit Suisse Forum St. Peter,
St. Peterstr. 19, 8001 Zürich.
Tuesday, April 8, 2003, 16.30 h
(Converium, General-Guisan-Quai 26.
8022 Zürich, 6th floor)
