Localized kernel‐based meshless method for pricing financial options underlying fractal transmission system
O. Nikan, Z. Avazzadeh, J. A. Tenreiro Machado
Abstract
The variation in the option pricing of the fractal transmission system is modelled by the time fractional Black–Scholes equation (TFBSE). This paper proposes an efficient local meshless method for the numerical simulation of the TFBSE. At the first step, a difference formula of L 1 type is employed to get a semi‐discrete algorithm in the temporal variable with a accuracy of order 2 − α in the case of smooth solutions, where 0 < α ≤ 1 is the fractional‐order derivative. At the second step, a localized radial basis function finite difference is adopted to derive a full‐discrete scheme. Moreover, the unconditional stability and convergence of the proposed method are analyzed based on energy norm. The exact expressions for the weights of the first and second derivatives are used by imposing a multiquadric function generated by finite difference. The proposed technique produces linear systems with tridiagonal and diagonal matrices. Numerical experiments highlight the performance of the method.