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Abstract To quickly solve the fractional Black–Scholes (B–S) equation in the option pricing problems, in this paper, we construct pure alternative segment explicit–implicit (PASE-I) and pure alternative segment implicit–explicit (PASI-E) difference schemes for time-space fractional B–S equation. It is a kind of intrinsic parallel difference schemes constructed on the basis of classic explicit scheme and classic implicit scheme combined with alternate segmentation technique. PASE-I and PASI-E schemes are analyzed to be unconditionally stable, convergent with second-order spatial accuracy and (2−α) $(2-\alpha)$th-order time accuracy, and they have a unique solution. The numerical experiments show that the two schemes have obvious parallel computing properties, and the computation time is greatly improved compared to Crank–Nicolson (C–N) scheme. The PASE-I and PASI-E intrinsic parallel difference methods are efficient to solve the time-space fractional B–S equation.

To quickly solve the fractional Black-Scholes (B-S) equation in the option pricing problems, in this paper, we construct pure alternative segment explicit-implicit (PASE-I) and pure alternative segment implicit-explicit (PASI-E) difference schemes for time-space fractional B-S equation. It is a kind of intrinsic parallel difference schemes constructed on the basis of classic explicit scheme and classic implicit scheme combined with alternate segmentation technique. PASE-I and PASI-E schemes are analyzed to be unconditionally stable, convergent with second-order spatial accuracy and (2−α)th-order time accuracy, and they have a unique solution. The numerical experiments show that the two schemes have obvious parallel computing properties, and the computation time is greatly improved compared to Crank-Nicolson (C-N) scheme. The PASE-I and PASI-E intrinsic parallel difference methods are efficient to solve the time-space fractional B-S equation. [ABSTRACT FROM AUTHOR]