Modeling of financial processes with a space-time fractional diffusion equation of varying order

Abstract In this paper, a new model for financial processes in form of a space-time fractional diffusion equation of varying order is introduced, analyzed, and applied for some financial data. While the orders of the spatial and temporal derivatives of this equation can vary on different time intervals, their ratio remains constant and thus the global scaling properties of its solutions are conserved. In this way, the model covers both a possible complex short-term behavior of the financial processes and their long-term dynamics determined by its characteristic time-independent scaling exponent. As an application, we consider the option pricing and describe how it can be modeled by the space-time fractional diffusion equation of varying order. In particular, the real option prices of index S&P 500 traded in November 2008 are analyzed in the framework of our model and the results are compared with the predictions made by other option pricing models.

[1]  J. Nolan,et al.  Maximum likelihood estimation and diagnostics for stable distributions , 2001 .

[2]  H. Kleinert.  Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets , 2006 .

[3]  佐藤 健一.  Lévy processes and infinitely divisible distributions , 2013 .

[4]  R. Gorenflo,et al.  AN OPERATIONAL METHOD FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS WITH THE CAPUTO DERIVATIVES , 1999 .

[5]  K. Chung,et al.  Limit Distributions for Sums of Independent Random Variables. , 1955 .

[6]  G. H. Erjaee,et al.  Examples of Analytical Solutions by Means of Mittag-Leffler Function of Fractional Black-Scholes Option Pricing Equation , 2015 .

[7]  Y. Chen,et al.  Variable-order fractional differential operators in anomalous diffusion modeling , 2009 .

[8]  Nikolai N. Leonenko,et al.  Fractional Skellam processes with applications to finance , 2014 .

[9]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[10]  Yuri Luchko,et al.  A New Fractional Calculus Model for the Two-dimensional Anomalous Diffusion and its Analysis , 2016 .

[11]  Yuri Luchko,et al.  Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process , 2016, Axioms.

[12]  Hagen Kleinert,et al.  Option Pricing Beyond Black-Scholes Based on Double-Fractional Diffusion , 2015, 1503.05655.

[13]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[14]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[15]  Hans U. Gerber,et al.  Option pricing by Esscher transforms. , 1995 .

[16]  H. Kleinert,et al.  Green function of the double-fractional Fokker-Planck equation: path integral and stochastic differential equations. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Xavier Gabaix,et al.  Power Laws in Economics and Finance , 2009 .

[18]  Yuri V. Vassilevski,et al.  Methods of blood flow modelling , 2016 .

[19]  F. Mainardi,et al.  The fundamental solution of the space-time fractional diffusion equation , 2007, cond-mat/0702419.

[20]  Yuri Luchko.  Multi-dimensional fractional wave equation and some properties of its fundamental solution , 2013, 1311.5920.

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

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

[23]  Laurent E. Calvet,et al.  Multifractal Volatility: Theory, Forecasting, and Pricing , 2008 .

[24]  Yuri Luchko,et al.  Wave-diffusion dualism of the neutral-fractional processes , 2015, J. Comput. Phys..

[25]  Paolo Grigolini,et al.  Scaling detection in time series: diffusion entropy analysis. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

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

[28]  Hagen Kleinert,et al.  Path Integrals in Quantum Mechanics , 1995 .

[29]  Rosario N. Mantegna,et al.  Book Review: An Introduction to Econophysics, Correlations, and Complexity in Finance, N. Rosario, H. Mantegna, and H. E. Stanley, Cambridge University Press, Cambridge, 2000. , 2000 .

[30]  Fawang Liu,et al.  Numerical techniques for the variable order time fractional diffusion equation , 2012, Appl. Math. Comput..

[31]  B. Mandelbrot.  The Variation of Certain Speculative Prices , 1963 .

[32]  Yuri Luchko.  Fractional wave equation and damped waves , 2012, 1205.1199.

[33]  S. Howison,et al.  Option Pricing with Levy-Stable Processes , 2004 .

[34]  Petr Jizba,et al.  Multifractal diffusion entropy analysis: Optimal bin width of probability histograms , 2014, 1401.3316.