Space Mapping in Computational Finance

My name is Anna Clevenhaus and I am a PhD student in the Applied and Computational Mathematics (ACM) group at the University of Wuppertal, where I also received my Bachelor’s and Master’s degrees. My main research interest is the improvement of numerical methods, especially the improvement of computational efficiency.

Why I became a PhD student
At the beginning of my studies I wanted to become a teacher for mathematics and chemistry, and during my bachelor studies my interest in solving mathematical problems and analytical thinking deepened. In addition, I took my first programming courses, which strengthened my new interest in applied mathematics. This is the reason why I chose the international program Computational Simulation in Science with a focus on Computational Finance. Fortunately, my PhD will allow me to deepen and broaden my knowledge, as there is always more to learn and explore.

The journey of my PhD
I started my study journey with a focus on Sparse Grids combined with American Option Pricing problems [1], followed by the integration of the Parareal algorithm within a financial setup [2]. Both numerical methods, the Sparse Grid approach and the Parareal algorithm, are hierarchical methods. The Sparse Grid approach considers several hierarchical spatial grids, while the Parareal algorithm is a temporal multi-grid approach with two layers. My Ph.D. thesis is a hierarchical approach to parameter calibration of the Heston model, where we introduce the space-mapping approach to the financial world.

In this project I was able to extend my knowledge of optimization by working with our colleague Claudia Totzeck. Within the Space-Mapping approach, one considers a coarse model to get a first idea of the optimal parameter set. It is therefore based on a model hierarchy. As a first step, we derived a gradient based method for parameter calibration for the Heston model [3], which is the core of Space-Mapping. The results for the parameter calibration of the Heston model are remarkable, as the figure shows the cost function improvement per iteration for 5 different randomly generated test sets. Thus, I am very excited about the integration of this method within the Space-Mapping approach as the next and final step.

Best regards.
M.Sc. Anna Clevenhaus

References:

[1] A. Clevenhaus, M. Ehrhardt and M. Günther,
An ADI Sparse Grid Method for the Efficient Pricing of American Options under the
Heston Model, Adv. Appl. Math. Mech. 13.6 (2021), pp. 1384-1397.

[2] A. Clevenhaus, M. Ehrhardt, and M. Günther,
The Parareal Algorithm and the Sparse Grid Combination Technique in the Application of the
Heston model, Progress in Industrial Mathematics at ECMI 2021.
Edited by M. Ehrhardt and M. Günther. Cham: Springer International Publishing, 2022, pp. 477-483.

[3] A. Clevenhaus, C. Totzeck, and M. Ehrhardt,
A gradient based calibration method for the Heston model,
IMACM Preprint 23/16 (Oct. 2023).