Preconditioned iterative methods for fractional diffusion models in finance

Most recent qualitative models for financial assets assume that the dynamics of underlying equity prices follows a jump or Levy process. It has been evident that some most intricate characteristics of such dynamics can be captured by CGMY and KoBoL procedures. The prices of financial derivatives with such models satisfy fractional partial differential equations or partial integro-differential equations. This study focuses at aforementioned fractional equations and discretizes them via a monotone Crank–Nicolson procedure. A spatial extrapolation strategy is introduced to ensure an overall second-order accuracy in approximations. Preconditioned conjugate gradient normal residual methods are incorporated for solving resulted linear systems. Numerical examples are given to illustrate the accuracy and efficiency of the novel computational approaches implemented. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1382–1395, 2015

[1]  B. Kleefeld,et al.  Solving complex PDE systems for pricing American options with regime‐switching by efficient exponential time differencing schemes , 2013 .

[2]  Koponen,et al.  Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  Diego del-Castillo-Negrete,et al.  Fractional diffusion models of nonlocal transport , 2006 .

[4]  M. Meerschaert,et al.  Finite difference approximations for two-sided space-fractional partial differential equations , 2006 .

[5]  Ebrahim Momoniat,et al.  A comparison of numerical solutions of fractional diffusion models in finance , 2009 .

[6]  Jesper Andreasen,et al.  Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Pricing , 1999 .

[7]  R. Chan,et al.  An Introduction to Iterative Toeplitz Solvers , 2007 .

[8]  M. Yousuf,et al.  An efficient ETD method for pricing American options under stochastic volatility with nonsmooth payoffs , 2013 .

[9]  K. I. '. Hout,et al.  ADI finite difference schemes for option pricing in the Heston model with correlation , 2008, 0811.3427.

[10]  R. C. Merton,et al.  Theory of Rational Option Pricing , 2015, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.

[11]  M. Yor,et al.  The Fine Structure of Asset Retums : An Empirical Investigation ' , 2006 .

[12]  Mark M. Meerschaert,et al.  A second-order accurate numerical approximation for the fractional diffusion equation , 2006, J. Comput. Phys..

[13]  X. Li,et al.  Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation , 2010 .

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

[15]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[16]  Hong Wang,et al.  A direct O(N log2 N) finite difference method for fractional diffusion equations , 2010, J. Comput. Phys..

[17]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[18]  I. Podlubny.  Fractional differential equations , 1998 .

[19]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[20]  W. Schoutens.  Lévy Processes in Finance: Pricing Financial Derivatives , 2003 .

[21]  Xiao-Qing Jin,et al.  Preconditioned iterative methods for fractional diffusion equation , 2014, J. Comput. Phys..

[22]  R. Cont,et al.  Financial Modelling with Jump Processes , 2003 .

[23]  Q. Sheng,et al.  A fully adaptive approximation for quenching‐type reaction‐diffusion equations over circular domains , 2014 .