# 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).