A Fast Compact Difference Scheme With Non-uniform Time Steps For The Tempered Time-fractional Black-Scholes Model

Black-Scholes(B-S)equation has been generalized as a tempered time-fraction B-S equation in recent years,which has become an important mathematical model in financial engineering.In this paper,we transform the tempered time-fraction B-S model into a special time-fractional diffusion response equation by means of exponential transformation technique.Then we discretized this special time-fractional diffusion response equation into a compact finite difference system,and studied its stability and convergence.In the direction of space,the compact operator is used to discrete the spatial derivative,and the difference scheme reaches the fourth order precision.In time direction,in order to overcome the initial singularity of the solution,the L1 formula under non-uniform time step is used to approximate the fractional derivative of tempered Caputo,so that the difference scheme can reach the min {γα,2-α} order precision in time,where the γ,α are grid distortion index of non-uniform time and the fractional derivative of tempered time order respectively.In addition,exponential and approximation methods are used to reduce the algorithm storage of kernel function in Caputo fractional derivative and reduce the computing time.In the end,three numerical examples are given to prove the effectiveness of the proposed method in fast time calculation and high order approximation in space.