The German Academic Exchange Service (DAAD) supports four 2-year projects by Thomas Kruse, Long Teng and Matthias Ehrhardt (Chair of Applied and Computational Mathematics, Wuppertal) as part of its project-based exchange program with France, Hong Kong, Portugal and Slovakia. This type of project funding is intended in particular to strengthen the scientific exchange between the two research institutions and to give young scientists the opportunity to gain international research experience through research stays at the partner institution.
The project partner in France is Alexandre Popier’s group at the Laboratoire Manceau de Mathématiques at the University of Le Mans. The funded project deals with the development of risk management strategies in illiquid financial markets. In particular, we will study the impact of stochastic liquidity on optimal trade execution strategies and quantify the associated price risks. Backward stochastic differential equations (BSDEs) have proven to be a powerful mathematical tool in this context. In the project, the resulting equations are checked for well-posedness and efficient numerical methods are developed to approximate their solutions.
In the joint project with the Chinese University of Hong Kong, we investigate the existence of optimal solution pairs for selected high-dimensional control problems in mathematical finance, such as the optimal investment-consumption problem, superreplication with portfolio constraints, and American options. We derive the necessary conditions for for the existence of solutions to the coupled forward-backward stochastic differential equations (FBSDEs) arising from these control problems. Furthermore, in order to develop a more effective solver for these control problems, in particular for the coupled FBSDEs they generate, we will design a better architecture of the neural network, and define a novel loss function that also incorporates information from the control problems. A rigorous error analysis of the proposed method will be performed.
The project partner of “PRISEMA – Pricing of Financial Instruments in Emission Markets” is João Guerra from ISEG, Lisbon. The aim of this project is to develop new models and methods for the and renewable energy certificates. We will focus on stochastic structural models based on the underlying economic factors that determine the price of carbon credits. Our goal is to derive a FBSDE for the allowance pricing process and solve the associated partial differential equations (PDEs) numerically. We will extend the basic model for risk-neutral pricing of carbon emission allowances by considering more general processes for the electricity demand process and for the processes modeling cumulative emissions and the interaction between the electricity and emission markets.
Finally, in the bilateral German-Slovak project “BraWu – Bratislava-Wuppertal research group on mathematical modeling in finance“, the partner is Daniel Sevcovic from the Comenius University in Bratislava. The aim of this project is to propose some generalizations/modifications of structural models and numerical methods previously presented in the literature for risk-neutral pricing of financial instruments in emission markets. For the proposed structural models, our goals are to derive a FBSDE for the price process of emission allowances and to solve numerically the associated nonlinear partial differential equations for the allowance price and for the prices of financial derivatives in the emission market.