Overlapping Multigrid Methods as an Efficient Approach for Solving the Black-Scholes Equation

• Autorzy: Michał Bernardelli
• Języki publikacji: EN

Abstrakty

EN

In this paper the modification of a two-level multigrid method by allowing an overlap between adjacent subdomains and its application to a one-dimensional Black-Scholes equation is described. The method is based on the finite-difference schema known as implicit Euler. Numerical experi10 ments confirm the superiority of the proposed method in relation to the clas11 sic multigrid method in form of shortening computation time, memory sav12 ings and ease of parallelization. The comparison shows the advantages of overlapping grids vs method without them, mainly due to improved accu14 racy of the solution.(original abstract)

Słowa kluczowe

EN

Black-Scholes model; Mathematical economics

PL

Model Blacka-Scholesa; Ekonomia matematyczna

• Czasopismo: Metody Ilościowe w Badaniach Ekonomicznych / Szkoła Główna Gospodarstwa Wiejskiego
• Rocznik: 2015
• Tom: 16(XVI)
• Numer: nr 1
• Strony: 25--36

Twórcy

autor

Michał Bernardelli

• Szkoła Główna Handlowa w Warszawie

Bibliografia

• Bernardelli M. (2008) Metody Dirichleta-Neumanna równoległego rozwiązywania dyskretyzacji zagadnień eliptycznych, PhD thesis, Warsaw University.
• Berridge S., Schumacher H. (2002) An Irregular Grid Method for Solving High- Dimensional Problems in Finance, Lecture Notes in Computer Science, Vol. 2330, 2002, pp. 510-519.
• Bhowmik S. K., Stolk C. C. (2011) Preconditioners based on windowed Fourier frames applied to elliptic partial differential equations, Journal of Pseudo-Differential Operators and Applications, 2(3) pp. 317-342.
• Chiorean I. (2005) Parallel algorithm for solving the Black-Scholes equation, Kragujevac J. Math. 27, pp. 91-100.
• Fries Ch. (2007), Mathematical Finance. Theory, modeling, implementation, Wiley.
• Haug E. G. (2007) The Complete Guide to Option Pricing Formulas, McGraw-Hill.
• Henshaw W. D. (2005) On Multigrid for Overlapping Grids, SIAM J. Sci. Comput., 26(5), pp. 1547-1572.
• Hull J. C. (2006) Futures, Options and Other Derivatives, Sixth Edition, Prentice Hall.
• Knabner P., Angerman L. (2003) Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer.
• Pascucci A. (2011) PDE and Martingale Methods in Option Pricing, Springer.
• Saad Y. (2003) Iterative Methods For Sparse Linear Systems, Second Edition, SIAM.
• Sauer T. (2012) Numerical Solution of Stochastic Differential Equations in Finance, Springer Handbooks of Computational Statistics 2012, pp. 529-550.
• Shapira Y. (2003) Matrix-Based Multigrid: Theory and Applications, Springer.
• Strang G. (2007) Computational Science and Engineering, Wellesley-Cambridge Press.
• Toselli A., Widlund O. (2005) Domain Decomposition Methods: Algorithms and Theory, Springer Series in Computational Mathematics, Vol. 34.
• Wesseling P. (2004) An Introduction to Multigrid Methods. John Wiley & Sons, Chichester, 1992. Reprinted by R.T. Edwards, Inc.
• Zhu Y., Cangellaris A. C. (2006) Multigrid finite element methods for electromagnetic field modeling. Wiley.
• Zienkiewicz O. C., Taylor R. L. Zhu J. Z. (2005) The finite element method: its basis and fundamentals, Elsevier Butterworth-Heinemann, Amsterdam, Boston.