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首页 > 过刊浏览>2016年第44卷第9期 >1458-1465. DOI:10.11908/j.issn.0253-374x.2016.09.022
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有限体积法定价跳扩散期权模型
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作者单位:

同济大学数学系,同济大学数学系,同济大学数学系

中图分类号:

O241.8

基金项目:

国家自然科学基金项目（填写项目编号）11271289,中央高校基本科研业务费专项资金, 云南省应用基础研究计划青年项目(2013FD045),云南省教育厅科学研究基金项目(2015Y443).


Finite Volume Methods for Pricing JumpDiffusion Option Model
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    摘要:

    考虑有限体积法求解Kou模型下美式跳扩散期权.基于线性有限元空间,构造了向后欧拉和CrankNicolson两种全离散有限体积格式,并采用简单高效的递推公式对偏微分积分方程中的积分项进行逼近.针对美式期权离散得到的线性互补问题(LCP),采用模超松弛迭代法(MSOR)进行求解,并证明了H+离散矩阵下算法的收敛性.数值实验表明,所构造的方法是高效而稳健的.

    Abstract:

    Finite volume methods are developed for pricing American options under Kou jumpdiffusion model. Based on a linear finite element space, both backward Euler and CrankNicolson full discrete finite volume schemes are constructed. For the approximation of the integral term in the partial integrodifferential equation (PIDE), an easytoimplement recursion formula is employed. Then we propose the modulusbased successive overrelaxation (MSOR) method for the resulting linear complementarity problems (LCPs). The H+ matrix property of the system matrix which guarantees the convergence of the MSOR method is analyzed. Numerical experiments confirm the efficiency and robustness of the proposed methods.

    参考文献
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    [3] Kou S G. A jump-diffusion model for option pricing [J]. Management Science, 2002, 48(8): 1086-1101.
    [4] Kou S G, Wang H. Option pricing under a double exponential jump diffusion model [J]. Management Science, 2004, 50(7): 1178-1198.
    [5] Tavella D, Randall C. Pricing financial instruments: The finite difference method [J]. John Wiley Sons, Chichester, UK, 2000.
    [6] Andersen L, Andreasen J. Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing [J]. Review of Derivatives Research, 2000, 4(3): 231-262.
    [7] 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.
    [8] d'Halluin Y, Forsyth P A, Vetzal K R. Robust numerical methods for contingent claims under jump diffusion processes [J]. IMA Journal of Numerical Analysis, 2005, 25(1): 87-112.
    [9] Toivanen J. Numerical valuation of European and American options under Kou's jump-diffusion model [J]. SIAM Journal on Scientific Computing, 2008, 30(4): 1949-1970.
    [10] Salmi S, Toivanen J. An iterative method for pricing American options under jump-diffusion model [J]. Applied Numerical Mathematics, 2011, 61(7): 821-831.
    [11] Zhang K, Wang S. Pricing options under jump diffusion processes with fitted finite volume method [J]. Applied Mathematics and Computation, 2008, 201(1): 398-413.
    [12] Zhang K, Wang S. A computational scheme for options under jump diffusion processes [J].International Journal of Numerical Analysis and Modeling, 2009, 6(1): 110-123.
    [13] Cryer C W. The solution of a quadratic programming using systematic overrelaxation [J]. SIAM Journal on Control and Optimization, 1971, 9(3): 385-392.
    [14] Ikonen S, Toivanen J. Operator Splitting Methods for American option pricing with stochastic volatility [J]. Numerische Mathematik, 2009, 113(2): 299-324.
    [15] 张凯. 美式期权定价—基于罚方法的金融计算 [M]. 经济科学出版社, 2012.Zhang K. Pricing American options—financial computation based on the penalty method [M]. Economic Science Press, 2012.
    [16] Murty K. Linear complementarity, linear and nonlinear programming [M]. Heldermann: Berlin, 1988.
    [17] Hadjidimos A, Tzoumas M. Nonstationary extrapolated modulus algorithms for the solution of the linear complementarity problem [J]. Linear Algebra and its Applications. 2009, 431(1): 197-210.
    [18] Dong J L, Jiang M Q. A modified modulus method for symmetric positive-definite linear complementarity problems [J]. Numerical Linear Algebra with Applications, 2009, 16(2): 129-143.
    [19] Bai Z Z. Modulus-based matrix splitting iteration methods for linear complementarity Problems [J]. Numerical Linear Algebra with Applications 2010, 17(6): 917-933.
    [20] Zheng N, Yin J F. Modulus-based successive overrelaxation method for pricing American options [J]. Journal of Applied Mathematics and Informatics, 2013, 31(5): 769-784.
    [21] Zheng N, Yin J F. Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem [J]. Numerical Algorithms, 2013, 64(2): 245-262.
    [22] Zheng N, Yin J F. Convergence of accelerated modulus-based matrix splitting iteration methods for linear complementarity problem with an matrix [J]. Journal of Computational and Applied Mathematics, 2014, 260: 281-293.
    [23] Kangro R, Nicolaides R. Far field boundary conditions for Black-Scholes equations [J]. SIAM Journal on Numerical Analysis, 2000, 38(4): 1357-1368.
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甘小艇,殷俊锋,李蕊.有限体积法定价跳扩散期权模型[J].同济大学学报(自然科学版),2016,44(9):1458~1465

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  • 收稿日期:2015-11-09
  • 最后修改日期:2016-06-07
  • 录用日期:2016-02-29
  • 在线发布日期: 2016-10-10
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