Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
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AMaMeF - Advanced Mathematical Methods for Finance

Stochastic Analysis, Stochastic Control, Nonlinear Differential Equations and Numerics: Applications to Option Pricing, Portfolio Optimization and Interest Rate Modeling

The AMaMeF project aims at the development and application of advanced mathematical tools in finance. It is funded by the European Science Foundation and many European national science foundations.

Prof. Dr. Michael Griebel and Dr. Thomas Gerstner from our research group are involved in the following projects of the AMaMeF program:

A detailed description of the two projects and of the AMaMeF program as a whole is given below.

About the AMaMeF program

Home page: http://www.iac.rm.cnr.it/amamef/

Project coordinators:

The AMaMeF project aims at the development and application of advanced mathematical tools in finance. Methods from stochastic analysis, deterministic and stochastic control theory, the theory of differential equations, functional analysis, mathematical statistics, numerical analysis and simulation, are playing an increasingly important role in the study of financial instruments and the very complex markets in which these instruments are traded. In particular, the problems associated with the pricing and hedging of derivatives, the implementation of large scale portfolio optimization methodologies, and the modeling of the dynamics of interest rates have inspired the development of sophisticated analytical tools and new mathematical methods of great importance for problems in financial economics. A good example is the novel use of Lévy processes as the basis for a more accurate description of the dynamics and statistical properties of asset prices. As a consequence of the great variety of techniques required for progress in the development of viable financial models and risk management tools, there is a serious need for a highly interdisciplinary approach to research in this area, an approach requiring expertise from a number of complementary areas of mathematics. The proposed project has two main overall goals. The first is the creation and reinforcement of relationships among European research teams in the fields of stochastic analysis, control theory, differential equations, and other relevant mathematical disciplines, with the purpose of undertaking and carrying out highly innovative interdisciplinary and interactive research in mathematical finance and its applications. The second is the cultivation and maintenance of strong and mutually reinforcing links with the financial services industry in the broadest sense, with a view to further enhancing the impact and influence of mathematical research on the financial industry. The existence in Europe of many well established research groups, all recognized centers of excellence, with many different domains of expertise, augurs well for the success of such a proposal scheme of collaboration. The presence of fifteen countries, and research groups led by many of the best international experts in the field, offers the opportunity within the scope of this proposal for (a) a very significant uplift in the overall level and visibility of European research, and (b) an effective and scientific response and way forward for addressing the many new European financial challenges.

Project "Sparse grid discretizations for jump diffusion processes"

Jump diffusion or Levy processes gain more and more importance for the modeling of stock or interest rate prices. One central problem is here to determine the prices of financial derivatives which have as underlying an asset which is governed by such a price dynamics. However, the valuation of such derivatives leads in general to high-dimensional integro-differential equations. Since these equations cannot be solved in closed form, numerical methods have to be employed. The focus of our research are so-called sparse grids which have been shown to be very efficient for the solution of high-dimensional partial differential equations and integration problems.

Project "Adaptive numerical valuation methods for Bermudean options"

Bermudean options allow the holder to exercise the option at a finite set of times until the expiration date. In contrast, European options allow only an exercise at the expiration date while American options allow an exercise at any time before the expiration date. The accurate numerical valuation of such options is difficult for the martingale approach as well as PDE-based approaches since the resulting discontinuities at the exercise times lead to a severe loss of accuracy in all standard methods. Our research aims towards the construction of iterative adaptive refinement methods which try to find and resolve these a priori unknown discontinuities efficiently.