Meshfree Approximation for Multi-Asset European and American Option Problems

Area 01 – Scienze matematiche e informatiche

This work is intended as a study of the applicability of the meshless approach of radial basis functions approximation to the solution of the well known Black-Scholes differential equation applied to European basket and multi-asset American option problems. First we price an European call option and an European basket call option by solving the Black-Scholes equation with two different radial basis functions (RBF) approaches, and with a finite centered differences method. Secondly, we study the applicability of meshfree approximation schemes for the solution of multi-asset American problems. We do this because these methods allow us to increase easily the number of assets without increasing drastically the computational cost. For American options, we do not know the exact solution of the Black-Scholes equation. Therefore, we consider as reference solution, the one obtained by a linearly implicit finite difference method in space. Then, as approximant, we compute the RBF solution. To integrate the time variable we use the θ-method. The resulting approximation compared with the reference solution turns out to be quite good since when the number of points grows, the Root Mean Square Error decreases. This comparison has been done for the cases of one and two underlying assets.Stefano De Marchi is associate professor in numerical analysis at the University of Padua. He is author and co-author of more than 70 scientific papers, many of them in approximation theory, multivariate polynomial interpolation, approximation by radial basis functions and recently also in rational interpolation. He is one of the discoverers of the so called "Padua points", which are the only set of quasi-optimal interpolation points explicitly known on the square, for polynomial interpolation of total degree. He is main editor of the open access journal Dolomites Research Notes on Approximation (DRNA), co-founder of the research group CAA (Constructive Approximation and Applications) between the Universities of Padua and Verona. He is also author of the books Funzioni Splines Univariate (Forum, Ed. Universitaria Udinese, 2001) and Appunti di Calcolo Numerico (Esculapio Ed. Bologna, 2011).Maddalena Mandarà received the degree in applied mathematics and the master degree in Quantitative Methods for the Finance, both at the University of Verona in 2008 and 2010, respectively. Currently she is Chief Quantitative Analyst at the DMDESF at the Centro Studi Alma Iura Srl. She is also co-founder of Numetrica.Anna Viero received the degree in applied mathematics in 2008 and the master degree in Quantitative Methods for the Finance in 2010, both at the University of Verona. Since 2010 she works as an expert for internal model validation for the rating at the Gruppo Banco Popolare.

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