A space-time fractional derivative model for European option pricing with transaction costs in fractal market ☆

Abstract From the point of view of fractional calculus and fractional differential equation, the work handles European option pricing problems with transaction costs in fractal market. Under the definition of the modified Riemman-Liouville fractional derivative, the pricing model based on a space-time fractional patrial differential equation is presented by the replicating portfolio, containing the Hurst exponent taken as the order of fractional derivative. And then, European call and put options are constructed and calculated by the enhanced technique of Adomian decomposition method under the finite difference frame. The fractional derivative model is finally tested by the data from the option market.

[1]  Lishang Jiang.  Mathematical Modeling and Methods of Option Pricing , 2005 .

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

[3]  Lina Song,et al.  A new improved Adomian decomposition method and its application to fractional differential equations , 2013 .

[4]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .

[5]  川口 光年,et al.  R.D.Richtmyer: Difference Methods for Initial-Value Problems. Interscience Pub. Inc. New York, 1957, xii+238頁, 15×23cm, \2,600. , 1958 .

[6]  Jun Wang,et al.  Option pricing of a bi-fractional Black-Merton-Scholes model with the Hurst exponent H in [1/2, 1] , 2010, Appl. Math. Lett..

[7]  Luca Vincenzo Ballestra,et al.  A NUMERICAL METHOD TO COMPUTE THE VOLATILITY OF THE FRACTIONAL BROWNIAN MOTION IMPLIED BY AMERICAN OPTIONS , 2013 .

[8]  Yan Zhang,et al.  Asian Option Pricing with Transaction Costs and Dividends under the Fractional Brownian Motion Model , 2014, J. Appl. Math..

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

[10]  S. Liao,et al.  Beyond Perturbation: Introduction to the Homotopy Analysis Method , 2003 .

[11]  Luca Vincenzo Ballestra,et al.  A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion , 2016 .

[12]  Daniel Sevcovic,et al.  On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile , 2005 .

[13]  Sample path properties of Gaussian processes , 2002 .

[14]  Xiao-Tian Wang.  Scaling and long range dependence in option pricing, IV: Pricing European options with transaction costs under the multifractional Black–Scholes model , 2010 .

[15]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[16]  Hsuan-Ku Liu,et al.  A closed-form approximation for the fractional Black-Scholes model with transaction costs , 2013, Comput. Math. Appl..

[17]  S. Liao.  Notes on the homotopy analysis method: Some definitions and theorems , 2009 .

[18]  Song-Ping Zhu,et al.  Analytically pricing European-style options under the modified Black-Scholes equation with a spatial-fractional derivative , 2014 .

[19]  Mark H. A. Davis,et al.  European option pricing with transaction costs , 1993 .

[20]  H. E. Hurst,et al.  Long-Term Storage Capacity of Reservoirs , 1951 .

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

[22]  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..

[23]  H. Leland..  Option Pricing and Replication with Transactions Costs , 1985 .

[24]  Tomas Björk,et al.  A note on Wick products and the fractional Black-Scholes model , 2005, Finance Stochastics.

[25]  G. Adomian.  A review of the decomposition method in applied mathematics , 1988 .

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

[27]  Lina Song,et al.  Approximate rational Jacobi elliptic function solutions of the fractional differential equations via the enhanced Adomian decomposition method , 2010 .

[28]  Guy Jumarie,et al.  Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations , 2008 .

[29]  Lin Sun.  Pricing currency options in the mixed fractional Brownian motion , 2013 .

[30]  P. Boyle,et al.  Option Replication in Discrete Time with Transaction Costs , 1992 .

[31]  S. Liao.  An optimal homotopy-analysis approach for strongly nonlinear differential equations , 2010 .

[32]  Jin-Rong Liang,et al.  Time-changed geometric fractional Brownian motion and option pricing with transaction costs , 2012 .

[33]  Guy Barles,et al.  Option pricing with transaction costs and a nonlinear Black-Scholes equation , 1998, Finance Stochastics.

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