Document Type : Research Article

Authors

1 University of Mazandaran

2 Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.

3 Faculty of Mathematical sciences, University of Mazandaran, Babolsar, Iran.

Abstract

‎The aim of this paper is to numerically price the European double barrier option by calculating the governing fractional Black-Scholes equation in illiquid markets‎. ‎Incorporating the price impact into the underlying asset dynamic‎, ‎which means that trading strategies affect the underlying price‎, ‎we consider markets with finite liquidity‎. ‎We survey both cases of first-order feedback and full feedback‎. ‎Asset evolution satisfies a stochastic differential equation with fractional noise‎, ‎which is more realistic in markets with statistical dependence‎. ‎Moreover‎, ‎the Sinc-collocation method is used to price the option‎. ‎Numerical experiments show that the results highly correspond to our expectation of illiquid markets‎.

Keywords

[1] N. H. Chan, C. T. Ng, Fractional constant elasticity of variance model, Lecture Notes-Monograph Series (2006) 149-164.
[2] S. Rostek, R. Schobel, A note on the use of fractional brownian motion for  nancial modeling, Economic Modelling 30 (2013) 30-35.
[3] K. J. Glover, P. W. Duck, D. P. Newton, On nonlinear models of markets with  nite liquidity: some cautionary notes, SIAM Journal on Applied Mathematics 70 (8) (2010) 32523271.
[4] G. Jumarie, Stock exchange fractional dynamics de ned as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations, Insurance: Mathematics and Economics 42 (1) (2008) 271287.
[5] S. Shari an, A. R. Soheili, A. Neisy, A Numerical solution for the new model of time fractional bond pricing: Using a multiquadric approximation method, Journal of Mathematics and Modeling in Finance, 2 (1) (2022), Doi.org/10.22054/jmmf.2022.68274.1056.
[6] M. Rezaei, A. Yazdanian, A. Ashra , S. Mahmoudi, Numerical pricing based on fractional Black-Scholes equation with time-dependent parameters under the CEV model: Double barrier options, Computers & Mathematics with Applications 90 (2021) 104-111.
[7] I. Podlubny, Fractional differential equations, mathematics in science and engineering (1999).
[8] F. Stenger, Numerical metods based on sinc and analytic functions (1993).
[9] H. Takahasi, M. Mori, Double exponential formulas for numerical integration, Publications of the Research Institute for Mathematical Sciences 9 (3) (1974) 721-741.
[10] T. Okayama, T. Matsuo, M. Sugihara, Error estimates with explicit constants for sinc approximation, sinc quadrature and sinc inde nite integration, Numerische Mathematik 124(2) (2013) 361-394.
[11] J. Lund, K. L. Bowers, Sinc methods for quadrature and differential equations, SIAM, (1992).
[12] A. Babaei, H. Jafari, S. Banihashemi, A numerical scheme to solve a class of two-dimensional nonlinear time-fractional diffusion equations of distributed order, Engineering with Computers (2020) 1-13.