求解时间分数阶B-S模型的高阶MQ拟插值方法

Abstract

Abstract:

In this paper, Quasi-interpolation is a kind of high accurate meshless approximation method with shape-preserving property, which is often used in engineering. Based on cubic multiquadric (MQ) quasi-interpolation method, this paper proposes a novel meshless numerical scheme for time fractional Black-Scholes (B-S) model. The stability and convergence of the method are given. Numerical simulation shows that the method has high-order accuracy and is easy to be implemented for the nonuniform knots.

Key words: Multiquadric quasi-interpolation, Black-Scholes model, Option pricing, Fractional order

CLC Number:

O174

Shengliang Zhang,Junxian Huang. A High Order MQ Quasi-Interpolation Method for Time Fractional Black-Scholes Model[J].Acta mathematica scientia,Series A, 2022, 42(5): 1496-1505.

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References 17

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α	M	L₂	rate (L₂)	L_∞	rate (L_∞)
0.7	4	6.732 × 10^?3		7.821 × 10^?3
	8	1.587 × 10^?3	2.09	1.784 × 10^?3	2.13
	16	2.986 × 10^?4	2.41	3.366 × 10^?4	2.42
	32	3.867 × 10^?5	2.95	4.586 × 10^?5	2.87
	64	3.225 × 10^?6	3.58	4.133 × 10^?6	3.47
0.3	4	6.851 × 10^?3		8.070 × 10^?3
	8	1.613 × 10^?3	2.09	1.882 × 10^?3	2.10
	16	3.055 × 10^?4	2.40	3.541 × 10^?4	2.41
	32	4.012 × 10^?5	2.93	4.630 × 10^?5	2.93
	64	3.396 × 10^?6	3.56	3.895 × 10^?6	3.57

α	N	L₂	rate (L₂)	L_∞	rate (L_∞)
0.7	10	3.124 × 10^?3		4.301 × 10^?3
	20	1.351 × 10^?3	1.27	1.847 × 10^?3	1.22
	40	5.240 × 10^?4	1.28	7.403 × 10^?4	1.31
	80	2.142 × 10^?4	1.32	2.854 × 10^?4	1.37
	160	8.096 × 10^?5	1.33	1.009 × 10^?4	1.48
0.3	10	3.670 × 10^?3		4.984 × 10^?3
	20	1.142 × 10^?3	1.68	1.583 × 10^?3	1.66
	40	3.560 × 10^?4	1.69	4.871 × 10^?4	1.69
	80	1.084 × 10^?4	1.71	1.494 × 10^?4	1.71
	160	3.330 × 10^?5	1.74	4.447 × 10^?5	1.74

c	α = 0.7		α = 0.3
	L₂	L_∞	L₂	L_∞
0.1	5.987 × 10^?5	8.344 × 10^?5	6.783 × 10^?5	9.045 × 10^?5
0.2	3.867 × 10^?5	4.586 × 10^?5	4.012 × 10^?5	4.630 × 10^?5
0.3	4.637 × 10^?5	6.690 × 10^?5	6.532 × 10^?5	8.046 × 10^?5
0.4	5.450 × 10^?5	7.098 × 10^?5	6.998 × 10^?5	8.993 × 10^?5
0.5	6.721 × 10^?5	9.873 × 10^?5	7.356 × 10^?5	9.567 × 10^?5
0.6	7.983 × 10^?5	1.032 × 10^?4	8.220 × 10^?5	1.356 × 10^?4

c	α=0.7		α=0.3
	L₂	L_∞	L₂	L_∞
0.73	9.332 × 10^?5	1.437 × 10^?4	1.230 × 10^?4	3.877 × 10^?4
0.63	2.173 × 10^?5	3.245 × 10^?5	3.875 × 10^?5	4.690 × 10^?5
0.53	2.356 × 10^?4	3.678 × 10^?4	3.256 × 10^?4	4.730 × 10^?4
0.43	5.889 × 10^?4	7.356 × 10^?4	6.709 × 10^?4	9.034 × 10^?4
0.33	1.853 × 10^?3	2.801 × 10^?3	2.089 × 10^?3	3.903 × 10^?3
0.23	4.743 × 10^?2	6.082 × 10^?2	6.044 × 10^?2	9.871 × 10^?2

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A High Order MQ Quasi-Interpolation Method for Time Fractional Black-Scholes Model

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