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.
László Gerencsér - SZTAKI
Tel: +36 1 4665 644