• 论文 •

1. 1. 广东金融学院 金融数学与统计学院, 广州 510521;
2. 湘潭大学数学与计算科学学院, 湘潭 411105;
3. 上海师范大学数理学院, 上海 200234
• 收稿日期:2021-12-04 出版日期:2022-11-14 发布日期:2022-11-08
• 通讯作者: 王晚生,Email:w.s.wang@163.com.
• 基金资助:
国家自然科学基金青年项目（12101141）、国家自然科学基金项目（12271367，12071403，11771060）、上海市科技计划项目（20JC1414200）和上海市自然科学基金项目（20ZR1441200）资助.

Chen Yingzi, Xiao Aiguo, Wang Wansheng. HIGH ORDER ADI SPLITTING SCHEME FOR STOCHASTIC VOLATILITY MODEL WITH JUMP[J]. Mathematica Numerica Sinica, 2022, 44(4): 466-480.

### HIGH ORDER ADI SPLITTING SCHEME FOR STOCHASTIC VOLATILITY MODEL WITH JUMP

Chen Yingzi1,2, Xiao Aiguo2, Wang Wansheng3

1. 1. School of financial mathematics and statistics, Guangdong University of Finance, Guangzhou 510521, China;
2. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China;
3. College of Mathematics & Science, Shanghai Normal University, Shanghai 200234, China
• Received:2021-12-04 Online:2022-11-14 Published:2022-11-08

For the partial integro-differential equations satisfied by the stochastic volatility model with jump, a new high-order alternating direction implicit (ADI) finite difference scheme is proposed. The model is a convection diffusion type initial boundary value problem with mixed derivatives and non constant coefficients. We combine different high-order spatial discretization with the time-step ADI splitting scheme proposed by Hundsdorfer and Verwer, and obtain an effective method with fourth-order accuracy in space and second-order accuracy in time, and analyze the stability of high-order ADI scheme by Fourier method. Finally, the higher-order convergence of the numerical method is verified by numerical experiments on the European put option pricing model.

MR(2010)主题分类:

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 [1] Bates D S. Jumps and stochastic volatility:Exchange rate processes implicit Deutsche mark options[J]. Review Financial Stud., 1996, 9:69-107.[2] Beam R M, Warming R F. Alternating Direction Implicit methods for parabolic equations with a mixed derivative[J]. SIAM J. Sci. Stat. Comput., 1980, 1(1):1-29.[3] Black F, Scholes M. The pricing of options and corporate liabilities[J]. J. Political Economy, 1973, 81:637-654.[4] Chen Y Z, Wang W S, Xiao A G. An efficient algorithm for options under Merton's jump-diffusion model on nonuniform grids[J]. Comput. Econ., 2019, 53:1565-1591.[5] Cont R, Voltchkova E. A finite difference scheme for option pricing in jump-diffusion and exponential Lévy models[J]. SIAM J. Numer. Anal., 2005, 43:1596-1626.[6] Düring B, Fournié M, Rigal A. High-order ADI schemes for convection-diffusion equations with mixed derivative terms[C]. In:Spectral and High Order Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering 95, Springer, Berlin, Heidelberg, 2013:217-226.[7] Düring B, Fournié M, Heuer C. High-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids[J]. J. Comput. Appl. Math., 2014:247-266.[8] Düring B, Miles J. High-order ADI scheme for option pricing in stochastic volatility models[J]. J. Comput. Appl. Math., 2017, 316:109-121.[9] Düring B, Pitkin A. High-order compact finite difference scheme for option pricing in stochastic volatility jump models[J]. J. Comput. Appl. Math., 2019, 35:201-217.[10] Fouque J P, Papanicolaou G, Sircar K R. Derivatives in financial markets with stochastic volatility[J], Cambridge University Press, Cambridge, UK, 2000.[11] Haentjens T, in't Hout K J. Alternating direction implicit finite difference schemes for the HestonHull-White partial differential equation[J]. J. Comput. Finance, 2012, 16:83-110.[12] Hendricks C, Heuer C, Ehrhardt M, Günther M. High-order ADI finite difference schemes for parabolic equations in the combination technique with applications in finance[J]. J. Comput. Appl. Math., 2017, 316:175-194.[13] in't Hout K J, Welfert B D. Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms[J]. Appl. Numer. Math., 2009, 59(3-4):677-692.[14] Huang J X, Zhu W L, Ruan X F. Option pricing using the fast Fourier transform under the double exponential jump model with stochastic volatility and stochastic intensity[J]. J. Comput. Appl. Math., 2014:152-159.[15] Hundsdorfer W. Accuracy and stability of splitting with stabilizing corrections[J]. Appl. Numer. Math. 2002, 42:213-233.[16] Hundsdorfer W, Verwer J G. Numerical solution of time-dependent advection-diffusion-reaction equations[M]. Springer Ser. Comput. Math., Springer-Verlag, Berlin, 2003.[17] Ikonen S, Toivanen J. Efficient numerical methods for pricing American options under stochastic volatility[J]. Numer. Methods Partial Differential Equations, 2008, 24(1):104-126.[18] Kirkby J L, Nguyen D. Efficient Asian option pricing under regime switching jump diffusions and stochastic volatility models[J]. Ann. Finance., 2020, 16:307-351.[19] Lee J, Lee Y. Stability of an implicit method to evaluate option prices under local volatility with jumps[J]. Appl. Numer. Math., 2015:20-30.[20] Merton R C. Theory of rational option pricing[J]. Bell J. Econ. Manag. Sci., 1973, 4(1):141-183.[21] Patel K S, Mehra M. Fourth order compact scheme for option prcing under Merton's and Kou's jump-diffusion models[J]. Int. J. Theor. Appl. Finance, 2019, 21(4):1-20.[22] Salmi S, Toivanen J. IMEX schemes for pricing options under jump-diffusion models[J]. Appl. Numer. Math., 2014, 84:33-45.[23] Salmi S, Toivanen J, von Sydow L. An IMEX-scheme for pricing options under stochastic volatility models with jumps[J]. SIAM J. Sci. Comp., 2014, 36(5), B817-B834.[24] von Sydow L, Toivanen J, Zhang C. Adaptive finite difference and IMEX time-stepping to price options under Bates model[J]. Int. J. Comput. Math. 2015, 92(12):2515-2529.[25] Tour G, Thakoor N, Tangman, Désiré Y, Bhuruth M. A high-order RBF-FD method for option pricing under regime-switching stochastic volatility models with jumps[J]. J. Comput. Sci., 2019, 35:25-43.[26] Wang W S, Chen Y Z, Fang H. On the variable two-step IMEX BDF method for parabolic integrodifferential equations with nonsmooth initial data arising in finance. SIAM J Numer. Anal., 2019, 57(3):1289-1317.[27] Zhu S P, Chen W T. A spectral-collocation method for pricing perpetual American puts with stochastic volatility[J]. Appl. Math. Comput., 2011, 217(22):9033-9040.[28] Zhu S P, Chen W T. A predictor-corrector scheme based on the ADI method for pricing American puts with stochastic volatility[J]. Comput. Math. Appl., 2011, 62(1):1-26.[29] Zhu W, Kopriva D A. A spectral element approximation to price European options with one asset and stochastic volatility[J]. J. Sci. Comput., 2010, 42(3):426-446.
 [1] 张铁,祝丹梅. 美式期权定价问题的变网格差分方法[J]. 计算数学, 2008, 30(4): 379-387. [2] 孙志忠,李雪玲. 反应扩散方程的紧交替方向差分格式[J]. 计算数学, 2005, 27(2): 209-224.