Fractional differential equations which can be regarded as a generalization of classical differential equations that involve non-integer order derivatives, have emerged as a powerful tool in modeling and describing many complex phenomena that may not captured by classical differential equations. These significant applications in a wide range of scientific and engineering field include anomalous diffusion process, viscoelastic materials, chemistry, economics and finance.
In the present paper two high-order compact difference schemes with graded meshes are proposed for solving the time-fractional Black-Scholes (B-S) equation. Firstly, the convection term by using an exponential transformation is eliminated. Second, by combining the sixth-order / eighth-order compact difference method with a temporal graded meshes – based trapezoidal formulation for the temporal integral term to obtain the fully discrete high-order compact difference scheme. The stability and convergence analysis of the both proposed schemes are studied by applying Fourier analysis. The effectiveness of the proposed schemes and the correctness of the theoretical results are verified by two numerical examples.
In the Introduction the numerical schemes for solving one of the fractional models applied in finance, so-called time-fractional Black-Scholes equation is presented and discussed.
In Section 2 how to obtain the fully discrete high-order compact difference scheme with temporal graded meshes is shown. For the purpose of numerical simulation, the time-fractional B-S equation is considered to be solved on a finite domain. In order to perform the derivation of the numerical schemes, here some assumptions are proposed. The sixth-order compact difference scheme is expressed in a matrix form.
In Section 3 the stability of the high-order compact difference schemes by using Fourier analysis is presented. It is proved that the proposed sixth-order and eighth-order compact difference schemes are stabile.
In Section 4 the error estimates of the both proposed schemes are presented. In Theorem 4.3 the error estimate for the sixth-order compact difference scheme is given. The same is made in Theorem 4.5 but for the eighth-order compact difference scheme.
In Section 5 two numerical examples are provided to verify the accuracy of the both high-order compact difference schemes besides, the numerical simulation for the time-fractional B-S model which describes the European option price are performed to illustrated the practicability of the proposed schemes. The results are graphically illustrated.
In Section 6 there is a small conclusion.