Adaptive option pricing based on a posteriori error estimates for fully discrete finite difference methods

被引:0
作者
Mao, Mengli [1 ,2 ]
Wang, Wansheng [2 ]
Tian, Tianhai [3 ]
Wang, Lehan [4 ]
机构
[1] Changsha Univ, Sch Math, Changsha 410022, Peoples R China
[2] Shanghai Normal Univ, Dept Math, Shanghai 200234, Peoples R China
[3] Monash Univ, Sch Math, Clayton, Vic, Australia
[4] Univ Sydney, Sch Math & Stat, Sydney, NSW 2006, Australia
基金
中国国家自然科学基金; 上海市自然科学基金;
关键词
Partial integro-differential equations; a posteriori error estimate; Implicit-explicit time discretization; Finite difference method; European option pricing; Jump-diffusion model; Time-space adaptivity; CRANK-NICOLSON METHOD; JUMP-DIFFUSION; PARABOLIC PROBLEMS; ELEMENT METHODS; EQUATIONS; SCHEMES; VALUATION; MODELS;
D O I
10.1016/j.cam.2024.116407
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
In this paper we study the a posteriori error estimates of numerical methods for parabolic partial integro-differential equations (PIDEs) which describe jump-diffusion option pricing models in finance. Due to the non-smoothness of the payoff function, the initial singularity of the solution may arise and therefore a posteriori error control and adaptivity are of the utmost importance in numerically solving such type of equations. The implicit-explicit (IMEX) Euler and IMEX Crank-Nicolson Adams-Bashforth methods are used for the time discretization and a second order finite difference method is used for the space discretization. By using the continuous and piecewise reconstructions, the upper and lower bounds of the a posteriori error estimates for the fully discrete finite difference methods are derived, and these error bounds depend on the discretization parameters and the data of the model problems only. Based on these a posteriori error estimates, we further develop a spatial-temporal adaptive algorithm. Numerical experiments are performed with uniform partitions and time-space adaptivity. The new adaptive algorithm significantly reduces the computational cost and provides effective error control for the numerical solutions.
引用
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页数:26
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共 43 条
  • [1] Achdou Y, 2005, FRONT APP M, P1
  • [2] Akrivis G, 2006, MATH COMPUT, V75, P511, DOI 10.1090/S0025-5718-05-01800-4
  • [3] The effect of mesh modification in time on the error control of fully discrete approximations for parabolic equations
    Baensch, E.
    Karakatsani, F.
    Makridakis, Ch.
    [J]. APPLIED NUMERICAL MATHEMATICS, 2013, 67 : 35 - 63
  • [4] A POSTERIORI ERROR CONTROL FOR FULLY DISCRETE CRANK-NICOLSON SCHEMES
    Baensch, E.
    Karakatsani, F.
    Makridakis, Ch.
    [J]. SIAM JOURNAL ON NUMERICAL ANALYSIS, 2012, 50 (06) : 2845 - 2872
  • [5] Bergam A, 2005, MATH COMPUT, V74, P1117, DOI 10.1090/S0025-5718-04-01697-7
  • [6] Operator splitting schemes for the two-asset Merton jump-diffusion model
    Boen, Lynn
    't Hout, Karel J. in
    [J]. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 2021, 387
  • [7] Brenner S.C., 2002, Texts in Applied Mathematics, Vsecond
  • [8] Implicit-explicit numerical schemes for jump-diffusion processes
    Briani, Maya
    Natalini, Roberto
    Russo, Giovanni
    [J]. CALCOLO, 2007, 44 (01) : 33 - 57
  • [9] A hybrid adaptive finite difference method powered by a posteriori error estimation technique
    Cao, Jun
    Zhou, Guofang
    Wang, Cheng
    Dong, Xinzhuang
    [J]. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 2014, 259 : 117 - 128
  • [10] A New Spectral Element Method for Pricing European Options Under the Black-Scholes and Merton Jump Diffusion Models
    Chen, Feng
    Shen, Jie
    Yu, Haijun
    [J]. JOURNAL OF SCIENTIFIC COMPUTING, 2012, 52 (03) : 499 - 518