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Compact Finite Difference Schemes of The Time Fractional Black-Scholes Model
Zhifeng Weng
Keywords:Time-fractional Black-Scholes equation, European option, exponential transformation, compact difference scheme.
      In this paper, three compact difference schemes for the time-fractional Black-Scholes model governing European option pricing are presented. Firstly, in order to obtain the fourth-order accuracy in space by applying the Pad\'{e} approximation, we eliminate the convection term of the B-S equation by an exponential transformation. Then the time fractional derivative is approximated by $L1$ formula, $L2 - 1_\sigma$ formula and $L1 - 2$ formula respectively, and three compact difference schemes with oders $O(\Delta t^{2-\alpha}+h ^4)$, $O(\Delta t^{2}+h ^4)$ and $O(\Delta t^{3-\alpha}+h ^4)$ are constructed. Finally, numerical example is carried out to verify the accuracy and effectiveness of proposed methods, and the comparisons of various schemes are given. The paper also provides numerical studies including the effect of fractional orders and the effect of different parameters on option price in time-fractional B-S model