Some Difference Numerical Methods Of Fractional Black-Scholes Equations

It is important in theoretical significance and practical value to study the numerical computation of fractional option pricing model (fractional Black-Scholes equation). Time fractional Black-Scholes equation and time-space fractional Black-Scholes equation are two kinds of fractional Black-Scholes equation. This paper researches numerical solution of these fractional option pricing models.θ-difference scheme and explicit-implicit scheme and implicit-explicit scheme are constructed for solving the time-fractional Black-Scholes equation. Implicit scheme and θ-difference scheme are constructed for solving the time-space fractional Black-Scholes equation. And the existence and uniqueness of solutions, the stability and convergence of these schemes are analyzed.For the time-fractional Black-Scholes equation, the θ-difference scheme is analyzed to be conditional stability, convergent, existence and uniqueness of solution. The explicit-implicit scheme and implicit-explicit scheme are unconditional stability difference schemes. For the time-space fractional Black-Scholes equation, the implicit scheme is unconditional stability and the θ-difference scheme is analyzed to be conditional stability, convergent, existence and uniqueness of solution.According to the numerical experiment, the explicit-implicit scheme and implicit-explicit scheme for time-fractional Black-Scholes equation have the same calculation and their computational efficiency (computing time) is about 30% higher than Crank-Nicolson(C-N) scheme.θ-difference scheme for solving time-space fractional option pricing model is more effective and the computational efficiency is higher than implicit scheme.Theoretical analysis and numerical experiments demonstrate θ-difference scheme, explicit-implicit scheme and implicit-explicit scheme in this paper are effective to solve fractional Black-Scholes equation. It affirms that fractional Black-Scholes equation is more in line with the actual financial market.