Circulant preconditioning technique for barrier options pricing under fractional diffusion models

In recent years, considerable literature has proposed the more general class of exponential Lévy processes as the underlying model for prices of financial quantities, which thus better explain many important empirical facts of financial markets. Finite moment log stable, Carr–Geman–Madan–Yor and KoBoL models are chosen from those above-mentioned models as the dynamics of underlying equity prices in this paper. With such models pricing barrier options, one kind of financial derivatives is transformed to solve specific fractional partial differential equations (FPDEs). This study focuses on numerically solving these FPDEs via the fully implicit scheme, with the shifted Grünwald approximation. The circulant preconditioned generalized minimal residual method which converges very fast with theoretical proof is incorporated for solving resultant linear systems. Numerical examples are given to demonstrate the effectiveness of the proposed preconditioner and show the accuracy of our method compared with that done by the Fourier cosine expansion method as a benchmark.

[1]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[2]  A. Wathen,et al.  Iterative Methods for Toeplitz Systems , 2005 .

[3]  S. Levendorskii,et al.  Non-Gaussian Merton-Black-Scholes theory , 2002 .

[4]  Ioannis Kyriakou,et al.  Monte Carlo Simulation of the CGMY Process and Option Pricing , 2013 .

[5]  M. Ng,et al.  Z-Transform and preconditioning techniques for option pricing , 2012 .

[6]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

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

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

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

[10]  Deng Ding,et al.  An efficient algorithm for Bermudan barrier option pricing , 2012 .

[11]  M. Benzi.  Preconditioning techniques for large linear systems: a survey , 2002 .

[12]  Cornelis W. Oosterlee,et al.  Pricing early-exercise and discrete barrier options by fourier-cosine series expansions , 2009, Numerische Mathematik.

[13]  Andrey Itkin,et al.  Efficient solution of structural default models with correlated jumps and mutual obligations , 2014, Int. J. Comput. Math..

[14]  Siu-Long Lei,et al.  A circulant preconditioner for fractional diffusion equations , 2013, J. Comput. Phys..

[15]  Andrey Itkin,et al.  Using Pseudo-Parabolic and Fractional Equations for Option Pricing in Jump Diffusion Models , 2010 .

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

[17]  Liming Feng,et al.  PRICING DISCRETELY MONITORED BARRIER OPTIONS AND DEFAULTABLE BONDS IN LÉVY PROCESS MODELS: A FAST HILBERT TRANSFORM APPROACH , 2008 .

[18]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[19]  Álvaro Cartea,et al.  Fractional Diffusion Models of Option Prices in Markets With Jumps , 2006 .

[20]  Andrey Itkin,et al.  Efficient Solution of Backward Jump-Diffusion PIDEs with Splitting and Matrix Exponentials , 2013, 1304.3159.

[21]  Raymond H. Chan,et al.  Conjugate Gradient Methods for Toeplitz Systems , 1996, SIAM Rev..

[22]  Hong Wang,et al.  A fast characteristic finite difference method for fractional advection–diffusion equations , 2011 .

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

[24]  Jesús Vigo-Aguiar,et al.  On smoothing of the Crank-Nicolson scheme and higher order schemes for pricing barrier options , 2007 .

[25]  Pingping Zeng,et al.  Pricing Barrier and Bermudan Style Options Under Time-Changed Lévy Processes: Fast Hilbert Transform Approach , 2014, SIAM J. Sci. Comput..

[26]  R. Schilling.  Financial Modelling with Jump Processes , 2005 .

[27]  G. Strang,et al.  Toeplitz equations by conjugate gradients with circulant preconditioner , 1989 .

[28]  Wen Chen,et al.  A finite difference method for pricing European and American options under a geometric Lévy process , 2014 .

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

[30]  Andrey Itkin,et al.  Splitting and Matrix Exponential Approach for Jump-Diffusion Models with Inverse Normal Gaussian, Hyperbolic and Meixner Jumps , 2014, Algorithmic Finance.

[31]  S. Kou,et al.  Numerical pricing of discrete barrier and lookback options via Laplace transforms , 2004 .

[32]  R. Varga.  Geršgorin And His Circles , 2004 .

[33]  Qin Sheng,et al.  Preconditioned iterative methods for fractional diffusion models in finance , 2015 .

[34]  P. Carr,et al.  The Finite Moment Log Stable Process and Option Pricing , 2003 .