第11 卷 第19 期 Vol.11 No.19 2018 年 10 月 October 2018 时间-空间分数阶 Black-Scholes 方程的 一类高效差分方法 李 玥,杨晓忠,孙淑珍 * (华北电力大学数理学院,北京 102206) 摘要:分数阶 Black-Scholes(B-S)方程的数值解法对金融衍生品定价研究发挥着显著的促进作用。针对时间- 空间分数阶 B-S 方程构造出显-隐(explicit-implicit,E-I)差分格式和隐-显(implicit-explicit,I-E)差分格式, 这类格式由古典显式格式和古典隐式格式相结合构造而成。理论分析证明了 E-I 和 I-E 格式解的存在唯一性、 无条件稳定性和收敛性。数值试验证实 E-I 和 I-E 格式具有相同的计算复杂度,在计算精度相近的条件下,其 计算时间比 Crank-Nicolson (C-N)格式减少约 33%,数值试验与理论分析结果一致。E-I 和 I-E 差分方法对求 解时间-空间分数阶 B-S 方程是高效可行的,同时也证明了分数阶 B-S 方程更符合实际金融市场。 关键词:计算数学;时间-空间分数阶 Black-Scholes(B-S)方程;显-隐差分格式和隐-显差分格式;稳定性; 收敛性;数值试验 中图分类号:O241.8 文献标识码:A 文章编号:1674-2850(2018)19-1902-12 A class of efficient difference methods for time-space fractional Black-Scholes equation LI Yue, YANG Xiaozhong, SUN Shuzhen (School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China) Abstract: The numerical solutions of the fractional Black-Scholes (B-S) equation play a significant role in the pricing study of many financial derivatives. This paper proposes a class of explicit-implicit (E-I) and implicit-explicit (I-E) difference schemes for time-space fractional B-S equation. The E-I and I-E schemes are combined by classic explicit scheme and classic implicit scheme. And their solutions are proved to be existing and unique, unconditionally stable and convergent by theoretical analysis. The numerical experiments demonstrate that the two schemes have the same computational complexity. Under the similar calculation 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. The E-I and I-E difference methods are efficient to solve the time-space fractional B-S equation and fractional B-S equation is more suitable for actual financial market. Key words: computational mathematics; time-space fractional Black-Scholes (B-S) equation; explicit-implicit (E-I) and implicit-explicit (I-E) difference schemes; stability; convergence; numerical experiments 0 引言 B-S 方程作为著名的金融数学基本方程之一,越来越受到经济学家和应用数学家的广泛关注。经典 的 B-S 期权定价公式是在一系列严苛的假设下获得的,为使理论价格更加符合实际报价,需要适当放松 基金项目:国家自然科学基金(11371135) 作者简介:李玥(1994—),女,硕士研究生,主要研究方向:分数阶偏微分方程的数值解法 通信联系人:杨晓忠,教授,主要研究方向:计算数学与科学工程计算. E-mail: [email protected]
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A class of efficient difference methods for time-space fractional Black-Scholes equation
LI Yue, YANG Xiaozhong, SUN Shuzhen
(School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China)
Abstract: The numerical solutions of the fractional Black-Scholes (B-S) equation play a significant role in the pricing study of many financial derivatives. This paper proposes a class of explicit-implicit (E-I) and implicit-explicit (I-E) difference schemes for time-space fractional B-S equation. The E-I and I-E schemes are combined by classic explicit scheme and classic implicit scheme. And their solutions are proved to be existing and unique, unconditionally stable and convergent by theoretical analysis. The numerical experiments demonstrate that the two schemes have the same computational complexity. Under the similar calculation 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. The E-I and I-E difference methods are efficient to solve the time-space fractional B-S equation and fractional B-S equation is more suitable for actual financial market. Key words: computational mathematics; time-space fractional Black-Scholes (B-S) equation; explicit-implicit (E-I) and implicit-explicit (I-E) difference schemes; stability; convergence; numerical experiments
[1] 姜礼尚,徐承龙,任学敏,等. 金融衍生产品定价的数学模型与案例分析[M]. 北京:高等教育出版社,2008. JIANG L S, XU C L, REN X M, et al. Mathematical model and case analysis for financial derivative pricing[M]. Beijing: Higher Education Press, 2008. (in Chinese)
[2] KWOK Y K. Mathematical models of financial derivatives[M]. 2nd ed. Berlin: Springer, 2008. [3] WYSS W. The fractional Black-Scholes equations[J]. Fractional Calculus and Applied Analysis, 2000, 3(1): 51-61. [4] CARTEA A, del-CASTILLO-NEGRETE D. Fractional diffusion models of option prices in markets with jumps[J]. Physica A:
Statistical Mechanics and its Applications, 2007, 374(2): 749-763. [5] JUMARIE G. Derivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time.
Application to Merton’s optimal portfolio[J]. Computers & Mathematics with Applications, 2010, 59(3): 1142-1164. [6] UCHAIKIN V V. Fractional derivatives for physicists and engineers. Volume I: background and theory[M]. Beijing: Higher
GUO B L, PU X K, HUANG F H. Fractional partial differential equations and their numerical solutions[M]. Beijiing: Science Press, 2011. (in Chinese)
[8] SONG L N, WANG W G. Solution of the fractional Black-Scholes option pricing model by finite difference method[J]. Abstract and Applied Analysis, 2013, 2013(1-2): 1-16.
[9] KUMAR S, KUMAR D, SINGH J. Numerical computation of fractional Black-Scholes equation arising in financial market[J]. Egyptian Journal of Basic and Applied Sciences, 2014, 1(3-4): 177-183.
[10] ZHANG H, LIU F, TURNER I, et al. Numerical solution of the time fractional Black-Scholes model governing European options[J]. Computers & Mathematics with Applications, 2016, 71(9): 1722-1783.
[11] 覃平阳,张晓丹. 空间-时间分数阶对流扩散方程的数值解法[J]. 计算数学,2008,30(3):305-310. QIN P Y, ZHANG X D. A numerical method for the space-time fractional convection-diffusion equation[J]. Mathematica Numerica Sinica, 2008, 30(3): 305-310. (in Chinese)
[12] 刘发旺,庄平辉,刘青霞. 分数阶偏微分方程数值方法及其应用[M]. 北京:科学出版社,2015. LIU F W, ZHUANG P H, LIU Q X. Numerical methods of fractional partial differential equations and applications[M]. Beijing: Science Press, 2015. (in Chinese)
[13] YANG X Z, WU L F, SUN S Z, et al. A universal difference method for time-space fractional Black-Scholes equation[J]. Advance in Difference Equations, 2016, 2016(1): 71.
[14] 孙志忠,高广花. 分数阶微分方程的有限差分方法[M]. 北京:科学出版社,2015. SUN Z Z, GAO G H. Finite difference methods for fractional differential equations[M]. Beijing: Science Press, 2015. (in Chinese)
[15] ZHUANG P, LIU F. Implicit difference approximation for the time fractional diffusion equation[J]. Journal of Applied Mathematics and Computing, 2006, 22(3): 87-99.
[16] CARR P, WU L R. Time-changed Levy processes and option pricing[J]. Journal of Financial Economics, 2014, 71(1): 113-141.