跳扩散过程下期权定价的数值方法

摘要/Abstract

摘要： 研究了跳扩散过程下期权价值所满足\,PIDE\,方程的数值计算方法. 利用四阶差分格式对空间离散, 引入四阶\,Lagrange\,插值多项式对边界进行延拓, 得到一个非齐次线性系统. 基于矩阵指数的\,$\mathrm{Pad\acute{e}}$\,逼近方法及其分数表示形式, 构建了一种高阶光滑\,Crank-Nicolson\,差分格式. 数值计算验证了该种方法的有效性, 讨论了跳跃强度对标准期权和障碍期权的影响. 与传统的\,Crank-Nicolson\,格式相比, 该格式很好地处理了在执行价格和障碍点附近数值震荡的问题. 该种方法亦可应用于一般具有非光滑边界的线性系统问题.

关键词: 期权, 跳扩散过程, 数值方法, $\mathrm{Pad\acute{e}}$\, 逼近, 光滑\, Crank-Nicolson\, 格式

Abstract: Numerical method for partial integro-differential equation (PIDE) resulting from option value under jump-diffusion process was studied. A non-homogeneous linear system was obtained by discretizing the spatial derivatives utilizing the fourth-order difference and extending boundary using fourth-order Lagrange interpolating polynomial. Based on $\mathrm{Pad\acute{e}}$\ approximations and partial fraction version of the matrix exponential, a high-order smoothing Crank-Nicolson scheme was constructed. Numerical calculation discussed the influence of jump intensity on vanilla option value and barrier option value, showed that the algorithm was efficient. Compared with classic Crank-Nicolson scheme, the numerical scheme avoided the spurious oscillation near the strike price and barrier value. The algorithm also can be used in the general linear boundary value problem which has non-smooth boundary.

Key words: option, jump-diffusion process, numerical method, $\mathrm{Pad\acute{e}}$ approximation, smoothing Crank-Nicolson scheme

中图分类号:

黎伟, 周圣武. 跳扩散过程下期权定价的数值方法[J]. 华东师范大学学报(自然科学版), 2012, 2012(4): 27-35.

LI Wei, ZHOU Sheng-wu. Numerical method for option pricing under jump-diffusion process[J]. Journal of East China Normal University(Natural Sc, 2012, 2012(4): 27-35.

参考文献

{1} BLACK F, SCHOLES M. The pricing of options and corporate liabilities[J]. Political Economy, 1973, 81: 637-659.
{2} MERTON R C. Option pricing when underlying stock returns are discontinuous[J]. Journal of Financial Economics, 1976, 3: 125-144.
{3} CONT R, VOLTCHKOVA E. A finite difference scheme for option pricing in jump diffusion and exponential Lévy models[J]. SIAM Journal on Numerical Analysis, 2005, 43: 1596-1626.
{4} CARR P, MAYO A. On the numerical evaluation of option prices in jump diffusion processes[J]. European Journal of Finance, 2007, 13: 353-372.
{5} TOIVANEN J, SALMI S. An iterative method for pricing American options under jump-diffusion models[J]. Applied Numerical Mathematics, 2011, 61: 821-831.
{6} TANGMAN D Y, GOPAUL A, BHURUTH M. Exponential time integration and Chebychev discretisation schemes for fast pricing of options[J]. Applied Numerical Mathematics, 2008, 58: 1309-1319.
{7} ZVAN R, VETZAL K R, FORSYTH P A. PDE methods for pricing barrier options[J]. Journal of Economic Dynamics and Control, 2000, 24: 1563-1590.
{8} WADE B A, KHALIQ A Q M, YOUSUF M, et al. On smoothing of the Crank-Nicolson scheme and higher order schemes for pricing barrier options[J]. Journal of Computational and Applied Mathematics, 2007, 204: 144-158.
{9} 甄莉君, 张兴永, 牛成虎, 黎伟. 挂钩黄金理财产品定价的数值方法[J]. 华东师范大学学报: 自然科学版, 2011, 5: 25-32.
{10} KWOK Y K. Mathematical models of financial derivatives[M]. Berlin: Springer-Verlag Press, 2008.
{11} THOMAS J W. Numerical partial differential equations: finite difference methods[M]. Berlin: Springer-Verlag Press, 1995.
{12} COMPANY R, $\mathrm{J\acute{O}DAR}$ L, PINTOS J R. Computing option pricing models under transaction costs[J]. Computers and Mathematics with Applications, 2010, 59: 651-662.
{13} SPIKE T L, SUN H W. Fourth order compact boundary value method for option pricing with jumps[J]. Advances in Applied Mathematics and Mechanics, 2009, 1(6): 845-861.
{14} 王明新. 算子半群与发展方程[M]. 北京: 科学出版社, 2006.
{15} 王仁宏, 朱功勤. 有理函数逼近及其应用[M]. 北京: 科学出版社, 2004.
{16} KHALIQ A Q M, MARTN-VAQUERO J, WADE B A, et al. Smoothing schemes for reaction-diffusion systems with nonsmooth data[J]. Journal of Computational and Applied Mathematics, 2009, 223: 374-386.
{17} MOHAMMAD S. Smoothing of Crank-Nicolson scheme for the two-dimensional diffusion with an integral condition[J]. Applied Mathematics and Computation, 2009, 214: 512-522.
{18} KHALIQ A Q M, VOSS D A, YOUSUF M. Pricing exotic options with L-stable\ $\mathrm{Pad\acute{e}}$ schemes[J]. Journal of Banking and Finance, 2007, 31: 3438-3461.
{19} YOUSUF M. On the class of high order time stepping schemes based on\ $\mathrm{Pad\acute{e}}$ approximations for the numerical solution of Burgers' equation [J]. Applied

Mathematics and Computation, 2008, 205: 442-453.
{20} KHALIQ A Q M, TWIZELL E H, VOSS D A. On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal\ $\mathrm{Pad\acute{e}}$ approximations[J]. Numerical Methods for Partial Differential Equations, 1993, 9: 107-116.
{21} MOHAMMAD S. Fourth order positively smoothed $\mathrm{Pad\acute{e}}$\ schemes for parabolic partial differential equations with nonlocal boundary conditions[J]. Applied Mathematical Sciences, 2010, 42: 2065-2080.
{22} YOUSUF M. Efficient L-stable method forparabolic problems with application to pricing American options under stochastic volatility[J]. Applied Mathematics and Computation, 2009, 213: 121-136.

[1]	纪宇, 何一璇, 吴国群, 吴敏. 基于Prony-like方法的第一类贝塞尔函数逼近[J]. 华东师范大学学报(自然科学版), 2019, 2019(6): 42-60.
[2]	岑苑君. 连续型美式分期付款看跌期权[J]. 华东师范大学学报(自然科学版), 2019, 2019(3): 24-34,62.
[3]	陈海珍, 周圣武, 孙祥艳. 混合分数布朗运动下的回望期权定价[J]. 华东师范大学学报(自然科学版), 2018, 2018(4): 47-58.
[4]	杨朝强. 一类特殊混合跳-扩散模型的欧式回望期权定价[J]. 华东师范大学学报(自然科学版), 2017, (4): 1-17.
[5]	耿延静, 周圣武. 混合分数跳-扩散模型下的亚式期权定价[J]. 华东师范大学学报(自然科学版), 2017, (3): 29-38.
[6]	李鑫, 张鲁明, 柴光颖, . 带波动算子的非线性Schrodinger方程的线性紧格式 (英)[J]. 华东师范大学学报(自然科学版), 2016, 2016(3): 1-8.
[7]	朱琳, 芮洪兴. 分数阶对流扩散方程的半加权有限差分格式[J]. 华东师范大学学报(自然科学版), 2015, 2015(6): 18-29.
[8]	岑苑君, 易法槐. 永久美式交叉期权[J]. 华东师范大学学报(自然科学版), 2014, 2014(3): 8-13.
[9]	吕利娟, 张兴永. 双跳-扩散过程下时间依赖型的脆弱期权定价[J]. 华东师范大学学报(自然科学版), 2014, 2014(1): 13-20, 26.
[10]	苏小囡, 王伟, 王文胜. Regime switching~模型下的幂式期权定价[J]. 华东师范大学学报(自然科学版), 2013, 2013(6): 32-39.
[11]	岑苑君. 美式领子期权定价分析[J]. 华东师范大学学报(自然科学版), 2013, 2013(6): 40-45, 56.
[12]	侯帅，潘伟. 未知信源数条件下基于合成空间谱的弱信号DOA估计方法[J]. 华东师范大学学报(自然科学版), 2013, 2013(6): 102-111.
[13]	顾惠, 张云秀. 次扩散BS模型下带交易费的期权定价[J]. 华东师范大学学报(自然科学版), 2012, 2012(5): 85-92.
[14]	彭斌, 彭菲. 跳分形过程下延展期权定价[J]. 华东师范大学学报(自然科学版), 2012, 2012(3): 30-40.
[15]	马维元, 刘华. 两边空间-时间分数阶扩散方程的加权有限差分格式[J]. 华东师范大学学报(自然科学版), 2012, 2012(3): 41-48,70.