美式期权定价的分数阶偏微分方程组及其数值离散方法

doi:10.12288/szjs.2014.3.229

数值计算与计算机应用 ›› 2014, Vol. 35 ›› Issue (3): 229-240.DOI: 10.12288/szjs.2014.3.229

• 论文 • 上一篇

美式期权定价的分数阶偏微分方程组及其数值离散方法

席钧^1,2, 曹建文³

1. 中国科学院软件研究所, 北京 100190;
2. 中国科学院大学, 北京 100049;
3. 中国科学院软件研究所, 北京 100190

收稿日期:2013-12-09 出版日期:2014-09-15 发布日期:2014-09-30
基金资助:
国家自然科学基金NSF （Grant No. 91230109）资助项目.

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席钧, 曹建文. 美式期权定价的分数阶偏微分方程组及其数值离散方法[J]. 数值计算与计算机应用, 2014, 35(3): 229-240.

Xi Jun, Cao Jianwen. FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS AND NUMERICAL DISCRETIZATION METHOD FOR PRICING AMERICAN OPTIONS[J]. Journal of Numerical Methods and Computer Applications, 2014, 35(3): 229-240.

FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS AND NUMERICAL DISCRETIZATION METHOD FOR PRICING AMERICAN OPTIONS

Xi Jun^1,2, Cao Jianwen³

1. Institute of Software, Chinese Academy of Sciences, Beijing 100190, China;
2. University of Chinese Academy of Sciences, Beijing 100049, China;
3. Institute of Software, Chinese Academy of Sciences, Beijing 100190, China

Received:2013-12-09 Online:2014-09-15 Published:2014-09-30

KOBOL、FMLS、CGMY等无限跳跃活动Lévy模型下，期权定价可以表达为分数阶偏微分方程. 欧式期权在部分情况下有解析表达式计算，而美式期权定价属于线性互补问题，在这些无限跳跃活动模型下表达为包含分数阶偏微分方程的方程组，其同欧式期权定价相比更加复杂，只能采用数值方法.

在Cartea导出的欧式期权方程基础上，本文利用线性互补理论推导出针对美式期权的分数阶偏微分方程组，利用罚方法将分数阶偏微分方程组转化为单一方程，采用Grünwald 公式对分数阶偏微分方程设计出相应的数值离散格式，利用有限差分方法得到了每个时间步上的线性方程系统，采用迭代算法进行了线性方程的求解，并进行了数值实验和结果分析，以此来证明分数阶偏微分方程组及其数值离散格式的有效性. 基于分数阶偏微分方程对美式期权定价方程组的推导和相应的数值离散格式，在当前的文献中未见报道.

关键词： 美式期权, 欧式期权, 分数阶偏微分方程, 线性互补问题, 数值离散

Under infinite jump activity models such as Kobol, FMLS and CGMY, the prices of financial derivatives, such as options, satisfy a fractional partial differential equation (FPDE). American options have an additional constraint for the value of the option, and due to this, they lead to linear complementarity problems (LCP). Thus, American options pricing are much more complicated than that of European Options. In this paper, based on the FPDE for pricing European options derived by Cartea, a method for pricing American options is proposed. In the frame of LCP, the FPDE is introduced to build a mathematical model for pricing American options. Then, the fractional parts are treated with Grunwald equation and a penalty method is employed to transform the LCP into linear equations at each time step in the scheme of finite difference. Finally,we present numerical tests which illustrate the effectiveness of the method.

Key words: American options, European options, fractional partial differential equation(FPDE), linear complementarity problem(LCP), numerical discretization

MR(2010)主题分类:

65C20

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[1] Bachelier L. Théorie de la spéculation[M]. Gauthier-Villars, 1900.

[2] Merton R C, Samuelson P A. Continuous-time finance[M]. 1992.

[3] Schoutens W. Lévy processes in finance: pricing financial derivatives[M]. 2003.

[4] Mandelbrot B B, Van Ness J W. Fractional Brownian motions, fractional noises and applications[J]. SIAM review, 1968, 10(4): 422-437.

[5] Merton R C. Option pricing when underlying stock returns are discontinuous[J]. Journal of financial economics, 1976, 3(1): 125-144.

[6] Kou S G. A jump-diffusion model for option pricing[J]. Management science, 2002, 48(8): 1086-1101.

[7] Mandelbrot B B. The variation of certain speculative prices[M]. Springer New York, 1997.

[8] Geman H. Pure jump Lévy processes for asset price modelling[J]. Journal of Banking and Finance, 2002, 26(7): 1297-1316.

[9] Carr P, Geman H, Madan D B, et al. The Fine Structure of Asset Returns: An Empirical Investigation[J]. The Journal of Business, 2002, 75(2): 305-333.

[10] Press S J. A compound events model for security prices[J]. Journal of business, 1967: 317-335.

[11] Koponen I. Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process[J]. Physical Review E, 1995, 52(1): 1197.

[12] Boyarchenko S I, Levendorskii S Z. Non-Gaussian Merton-Black-Scholes Theory[M]. Singapore: World Scientific, 2002.

[13] 严加安. 金融数学: 历史, 现状和展望[J]. 2000.

[14] 姜礼尚. 期权定价的数学模型和方法[M]. 高等教育出版社, 2008.

[15] 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.

[16] 郭柏灵, 蒲学科, 黄凤辉. 分数阶偏微分方程及其数值解[J]. 2011.

[17] 辛宝贵, 陈通, 刘艳芹. 一类分数阶混沌金融系统的复杂性演化研究[J]. 物理学报, 2011, 60(4): 797-802.

[18] Cartea A. Dynamic hedging of financial instruments when the underlying follows a non-Gaussian process[R]. Working paper, Birkbeck College, University of London, 2005.

[19] Scalas E, Gorenflo R, Mainardi F. Fractional calculus and continuous-time finance[J]. Physica A: Statistical Mechanics and its Applications, 2000, 284(1): 376-384.

[20] F. Mainardi, M. Raberto, R. Gorenflo, E. Scalas, Fractional calculus and continuous-time finance II: the waiting-time distribution[J]. Physica A, 2000, 287: 469-481.

[21] d'Halluin Y, Forsyth P A, Labahn G. A penalty method for American options with jump diffusion processes[J]. Numerische Mathematik, 2004, 97(2): 321-352.

[22] Ikonen S, Toivanen J. Operator splitting methods for American option pricing[J]. Applied mathematics letters, 2004, 17(7): 809-814.

[23] Lord R, Fang F, Bervoets F, et al. A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes[J]. SIAM Journal on Scientific Computing, 2008, 30(4): 1678-1705.

[24] Marom O, Momoniat E. A comparison of numerical solutions of fractional diffusion models in finance[J]. Nonlinear Analysis: Real World Applications, 2009, 10(6): 3435-3442.

[25] Meerschaert M M, Tadjeran C. Finite difference approximations for two-sided space-fractional partial differential equations[J]. Applied Numerical Mathematics, 2006, 56(1): 80-90.

[26] Tadjeran C, Meerschaert M M, Scheffler H P. A second-order accurate numerical approximation for the fractional diffusion equation[J]. Journal of Computational Physics, 2006, 213(1): 205-213.

[27] Podlubny I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198 of[J]. Mathematics in Science and Engineering, 1999.

[28] Merton R C. Theory of rational option pricing[M]. 1971.

[29] Salmi S, Toivanen J. An iterative method for pricing American options under jump-diffusion models[J]. Applied Numerical Mathematics, 2011, 61(7): 821-831.

[30] Carr P, Wu L. The finite moment log stable process and option pricing[J]. The Journal of Finance, 2003, 58(2): 753-778.

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美式期权定价的分数阶偏微分方程组及其数值离散方法

FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS AND NUMERICAL DISCRETIZATION METHOD FOR PRICING AMERICAN OPTIONS

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