A new operator splitting method for American options under fractional Black-Scholes models

Abstract A new operator splitting method is proposed for American options under time-fractional Black–Scholes models. The fractional linear complementarity problem is split into two easy sub-problems, with the leading coefficients separated from the convolution sum and matched through a general correction step. The method is implementation friendly in the sense that one can easily modify a fractional European solver to obtain the proposed method, since the correction step is decoupled and is trivial to solve. The method is validated through numerical experiments and demonstrated to be superior to the traditional approach. The paper also provides numerical studies including the effect of fractional orders and the comparison of fractional models.

[1]  Fawang Liu,et al.  Numerical solution of the time fractional Black-Scholes model governing European options , 2016, Comput. Math. Appl..

[2]  Devendra Kumar,et al.  Numerical computation of fractional Black–Scholes equation arising in financial market , 2014 .

[3]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[4]  Jari Toivanen,et al.  COMPONENTWISE SPLITTING METHODS FOR PRICING AMERICAN OPTIONS UNDER STOCHASTIC VOLATILITY , 2007 .

[5]  Jari Toivanen,et al.  Application of operator splitting methods in finance , 2015, 1504.01022.

[6]  Jari Toivanen,et al.  Operator splitting methods for American option pricing , 2004, Appl. Math. Lett..

[7]  N. Nematollahi,et al.  New Splitting Scheme for Pricing American Options Under the Heston Model , 2018 .

[8]  Younhee Lee,et al.  A Second-Order Tridiagonal Method for American Options under Jump-Diffusion Models , 2011, SIAM J. Sci. Comput..

[9]  G. Jumarie,et al.  Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results , 2006, Comput. Math. Appl..

[10]  B. Øksendal,et al.  FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE , 2003 .

[11]  Jorge Nocedal,et al.  On the solution of complementarity problems arising in American options pricing , 2010, Optim. Methods Softw..

[12]  Jari Toivanen,et al.  Operator splitting methods for pricing American options under stochastic volatility , 2009, Numerische Mathematik.

[13]  Jaewook Lee,et al.  Tridiagonal implicit method to evaluate European and American options under infinite activity Lévy models , 2013, J. Comput. Appl. Math..

[14]  Jian Huang,et al.  Numerical approximation of a time-fractional Black-Scholes equation , 2018, Comput. Math. Appl..

[15]  S. Orszag,et al.  High-order splitting methods for the incompressible Navier-Stokes equations , 1991 .

[16]  G. H. Erjaee,et al.  A New Version of Black–Scholes Equation Presented by Time-Fractional Derivative , 2018 .

[17]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[18]  Yang Xiaozhong,et al.  A universal difference method for time-space fractional Black-Scholes equation , 2016 .

[19]  Haiming Song,et al.  Projection and Contraction Method for the Valuation of American Options , 2015 .

[20]  Song-Ping Zhu,et al.  Analytically pricing double barrier options based on a time-fractional Black-Scholes equation , 2015, Comput. Math. Appl..

[21]  Jari Toivanen,et al.  A high-order front-tracking finite difference method for pricing American options under jump-diffusion models , 2010 .

[22]  Jari Toivanen,et al.  Reduced order models for pricing European and American options under stochastic volatility and jump-diffusion models , 2016, J. Comput. Sci..

[23]  Miglena N. Koleva,et al.  Numerical solution of time-fractional Black–Scholes equation , 2017 .

[24]  Diego A. Murio,et al.  Implicit finite difference approximation for time fractional diffusion equations , 2008, Comput. Math. Appl..

[25]  Willem Hundsdorfer,et al.  On Multistep Stabilizing Correction Splitting Methods with Applications to the Heston Model , 2018, SIAM J. Sci. Comput..

[26]  Rob H. De Staelen,et al.  Numerically pricing double barrier options in a time-fractional Black-Scholes model , 2017, Comput. Math. Appl..

[27]  R. Glowinski.  Finite element methods for incompressible viscous flow , 2003 .

[28]  L. Bachelier,et al.  Théorie de la spéculation , 1900 .

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

[30]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[31]  Yanli Zhou,et al.  Fractional Order Stochastic Differential Equation with Application in European Option Pricing , 2014 .

[32]  George Labahn,et al.  A penalty method for American options with jump diffusion processes , 2004, Numerische Mathematik.

[33]  B. Henry,et al.  The accuracy and stability of an implicit solution method for the fractional diffusion equation , 2005 .

[34]  Guy Jumarie,et al.  Derivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time. Application to Merton's optimal portfolio , 2010, Comput. Math. Appl..

[35]  Eduardo S. Schwartz,et al.  The Valuation of American Put Options , 1977 .

[36]  Haiming Song,et al.  Primal-Dual Active Set Method for American Lookback Put Option Pricing , 2017 .

[37]  W. Wyss,et al.  THE FRACTIONAL BLACK-SCHOLES EQUATION , 2000 .

[38]  Lina Song,et al.  Solution of the Fractional Black-Scholes Option Pricing Model by Finite Difference Method , 2013 .

[39]  Guy Jumarie,et al.  On the fractional solution of the equation f(x + y) = f(x)f(y) and its application to fractional Laplace's transform , 2012, Appl. Math. Comput..

[40]  C. Cryer.  The Solution of a Quadratic Programming Problem Using Systematic Overrelaxation , 1971 .

[41]  Younhee Lee.  Financial options pricing with regime-switching jump-diffusions , 2014, Comput. Math. Appl..