Fast and efficient numerical methods for an extended Black-Scholes model

We study fast and efficient numerical approximations of a partial integro-differential equation (PIDE) that arises in option pricing theory (financial problems) as well as in various scientific modeling and deal with two different topics. As a first part of this article, we rigorously study different preconditioned iterative schemes for the full discrete model equation, focusing in particular on the choice of a fast and efficient solver. We develop preconditioners using wavelet basis transformations and Fourier sine transformations to improve the convergence criteria of iterative solvers for the full discrete model. We implement a multigrid (MG) iterative method for the discrete model problem as well. In fact, we approximate the problem using a finite difference scheme, then implement a few preconditioned Krylov subspace methods as well as a MG method to speed up the computation. A series of numerical experiments demonstrates that the number of iterations required for convergence is small and independent of the system size. In this sense the methods are optimal. Then, in the second part in this study, we analyze the stability and the accuracy of two different one step schemes to approximate the model.

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  Rama Cont,et al.  A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models , 2005, SIAM J. Numer. Anal..

[3]  Filo Fiorani.  Option Pricing under the Variance Gamma Process , 2004 .

[4]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[5]  J. Medlock,et al.  Spreading disease: integro-differential equations old and new. , 2003, Mathematical biosciences.

[6]  Gene H. Golub,et al.  Matrix computations , 1983 .

[7]  K. Chen,et al.  Matrix preconditioning techniques and applications , 2005 .

[8]  Yixin Guo,et al.  Existence and stability of standing pulses in neural networks , 2003 .

[9]  Stphane Mallat,et al.  A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way , 2008 .

[10]  James S. Walker.  Fast Fourier Transforms , 1991 .

[11]  Karsten Urban,et al.  Wavelet Methods for Elliptic Partial Differential Equations , 2008 .

[12]  D. Duffy.  Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach , 2006 .

[13]  高等学校計算数学学報編輯委員会編,et al.  高等学校計算数学学報 = Numerical mathematics , 1979 .

[14]  Khosrow Maleknejad,et al.  A comparison of Fourier extrapolation methods for numerical solution of deconvolution , 2006, Appl. Math. Comput..

[15]  S. K. Bhowmik,et al.  Preconditioners based on windowed Fourier frames applied to elliptic partial differential equations , 2010, 1009.1925.

[16]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[17]  Coarsening in an integro-differential model of phase transitions , 2000, European Journal of Applied Mathematics.

[18]  G. Lord,et al.  Waves and bumps in neuronal networks with axo-dendritic synaptic interactions , 2003 .

[19]  Roberto Natalini,et al.  Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory , 2004, Numerische Mathematik.

[20]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[21]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[22]  S. K. Bhowmik.  Numerical approximation of a convolution model of · θ -neuron networks , 2011 .