Several Efficient Difference Methods For Fractional Black-Scholes Equations In Statistical Dynamics

The numerical solutions of the fractional Black-Scholes(B-S)models play a significant role in promoting the study of the pricing of many financial derivatives.This paper proposes two class of difference methods for time-space fractional B-S equation:explicit-implicit(E-I)scheme and implicit-explicit(I-E)scheme,pure alternative segment explicit-implicit(PASE-I)scheme and pure alternative segment implicit-explicit(PASI-E)scheme.E-I scheme and I-E scheme are combined by classic explicit scheme and classic implicit scheme.Both of them are analyzed to be unconditionally stable and convergent and their solutions are existing and unique.The numerical experiments prove that the two schemes have the same computational complexity.Under the same precision,they save about 33%of the computation time compared to Crank-Nicolson(C-N)scheme.The numerical experiments are consistent with the theoretical analysis.PASE-I scheme and PASI-E scheme are constructed on the basis of classic explicit scheme and classic implicit scheme combined with alternate piecewise technique.PASE-I and PASI-E schemes are analyzed to be unconditionally stable,convergent with second order spatial accuracy and 2-? order time accuracy.Their solutions are existing and unique.The numerical experiments prove that the two schemes have obvious parallel nature and the computation time is greatly improved compared to Crank-Nicolson(C-N)scheme.The paper compares and analyzes these three schemes:C-N scheme,E-I scheme and PASE-I scheme.The comparison shows that the intrinsic parallel difference scheme has more obvious advantages in terms of computational efficiency and calculation accuracy compared with the traditional serial E-I scheme and C-N scheme.Numerical examples also confirm that the E-I scheme and PASE-I scheme constructed by this paper are efficient and feasible for solving time-space fractional B-S equations and fractional B-S equation is more suitable for actual financial market.