Stochastic Systems in Financial Mathematics - Research activities
at SZTAKI
by László Gerencsér
Financial mathematics and mathematical methods in economy have
attracted a lot of attention within academia in Hungary in recent
years. The potentials of the new area has also been recognized
at the SZTAKI: an inter-laboratory virtual research group has
been established by the name ‘Financial Mathematics and Management’.
The participating laboratories are: Laboratory of Applied Mathematics,
Laboratory of Operations Research and Decision Systems and Laboratory
of Engineering and Management Intelligence. The participants have
committed themselves to carrying out research, among other things,
in the area of option pricing, economic time series and portfolio
analysis. This article gives a short overview of the activity
of the Stochastic Systems Research Group, Laboratory of Applied
Mathematics and the Laboratory of Operations Research and Decision
Systems in the stochastic aspects of financial mathematics.
Our activity in the area started with my discussions with Tomas
Björk (Department of Finance, Stockholm School of Economics) and
Andrea Gombani (CNR/LADSEB) in summer, 1996, while visiting Lorenzo
Finesso in CNR/LADSEB. A prime theme for these discussions was
financial mathematics that attracted many people working in stochastic
analysis both in Europe and the USA last years. To try to use
our specialized skills a formal procedure was initiated at the
SZTAKI to get a project in financial mathematics established.
The initiative was accepted and the inter-laboratory virtual research
group ‘Financial Mathematics and Management’ was established.
Our research efforts, in the stochastic aspects, are focused on
market incompleteness due to uncertainties such as poor volatility
estimates in modeling the stock-processes. Under too much modeling
uncertainties the market is incomplete, and replicating a contingent
claim requires a non-self-financing portfolio. We have analyzed
the path-wise add-on cost and used it in formulating a stochastic
programming problem which yields a performance index for any given
price on which the seller and buyer agree. This approach has been
motivated by my earlier research with Jorma Rissanen in the area
of stochastic complexity on the interaction of statistical uncertainty
and performance. The method is a result of my joint work with
György Michaletzky, head of department at the Eötvös Loránd University
(ELTE), Budapest, and a part-time researcher at the SZTAKI, an
international authority on stochastic realization theory, and
with Miklós Rásonyi, the youngest member of the Stochastic Systems
Research Group. To get a data-driven procedure we also consider
the analysis of financial data by using on-line statistical analysis,
including adaptive prediction and change-detection. Zsuzsanna
Vágó, member of Laboratory of Operations Research and Decision
Systems, has obtained a János Bolyai research scholarship for
three years to study these problems.
In addition to research, we have started an educational program.
First, we had set up a one-semester course on derivative pricing.
An adequate place for this course was the Department of Probability
Theory and Statistics at the Eötvös Loránd University, headed
by György Michaletzky.
A major thrust to our educational activity was a one-week thrilling
minicourse, 14-20 September, 1998, held by Tomas Björk, with the
title ‘Arbitrage pricing of derivative financial securities’.
The course attracted some 30 enthusiastic participants from industry
and academia. Taking the advantage of this visit, we restructured
our educational program and now we have a two-semester course,
including more material on interest rate theory.
We are looking forward to having our next minicourse in financial
mathematics to be held next September, with the title ‘Optimal
Portfolios - Risk and Return Control’, given by Ralf Korn, Department
of Mathematics, University of Kaiserslautern.
We have been in co-operation with Manfred Deistler, Technical
University, Wien, in the area of time-series analysis, especially
with respect to co-integration. A joint project with Youri Kabanov,
Department of Mathematics at Université de Franche-Comté, Besancon,
France, including problems of option pricing and hedging under
transaction costs is just under way. We also see risk-sensitive
control, an area that has been significantly enriched by Jan van
Schuppen, CWI, as a potentially useful tool for portfolio design
and an area for further cooperation. We are looking forward to
developing a co-operative project with the group on research theme
‘Mathematics of Finance’, CWI, headed by Hans Schumacher.
Please contact:
László Gerencsér - SZTAKI
Tel: +36 1 4665 644
E-mail: gerencser@sztaki.hu
