Numerical approximation and fast implementation to a generalized distributed-order time-fractional option pricing model

We present a fully-discrete finite element scheme to a generalized distributed-order time-fractional option pricing model, which adequately describes, e.g., the valuation of the European double barrier option. Due to the dependence of the density function on the stock price, the temporal discretization coefficients from the generalized distributed-order time-fractional derivative will be coupled with the inner product of the finite element method, which significantly complicates the analysis and traditional numerical analysis techniques do not apply. Novel techniques are developed to prove error estimates of this fully-discrete numerical scheme, which not only resolves the above difficulty, but indeed simplifies existing methods by avoiding the mathematical induction procedure. Based on the structure of the all-at-once coefficient matrix of the proposed numerical scheme, a fast divide and conquer algorithm is developed to reduce the computational cost of solving the numerical scheme from O(LNt2Nx) to O(LNtlogNtNx), where L, Nt and Nx refer to numbers of the degree of freedom of discretizations for the distributed-order integral, the spatial domain and the time period, respectively. Numerical experiments are performed to demonstrate the accuracy of the proposed numerical scheme and its applications in the valuation of the option price.