Lattice Boltzmann Model for Pricing American Lookback Option

doi:10.13438/j.cnki.jdzk.2025.02.003

Abstract

Abstract: The lattice Boltzmann model (LBM) is used to solve the pricing problem of the American lookback option.The domain truncation technique and the penalty method are used to transform the linear complementary model satisfied by the lookback option into a one-dimensional nonlinear parabolic problem on a bounded domain,and LBM is introduced for the transformed bounded domain problem.The macroscopic equation is recovered by selecting appropriate equilibrium distribution functions and amending functions.The numerical simulation results show that the option prices obtained by LBM agree well with the results derived from the binomial tree method and other methods,which verify the effectiveness of the model in solving the pricing problem of the American lookback option.

Key words: lattice Boltzmann model, American lookback option, linear complementarity problem, domain truncation technique, penalty method

ZHANG Yi, WU Fangfang, ZHANG Qi. Lattice Boltzmann Model for Pricing American Lookback Option[J]. Journal of Jishou University(Natural Sciences Edition), 2025, 46(2): 23-30.

References

[1] 姜礼尚.期权定价的数学模型和方法[M].2版.北京:高等教育出版社,2008:1.
[2] WANG S,YANG X Q,TEO K L.Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation[J].Journal of Optimization Theory and Applications,2006,129(2):227-254.
[3] CHERNOGOROVA TATIANA P,KOLEVA MIGLENA N,VALKOV RADOSLAV L.A Two-Grid Penalty Method for American Options[J].Computational and Applied Mathematics,2018,37(3):2381-2398.
[4] KALANTARI R,SHAHMORAD S.A Stable and Convergent Finite Difference Method for Fractional Black-Scholes Model of American Put Option Pricing[J].Computational Economics,2019,53:191-205.
[5] ZHANG Qi,SONG Haiming,HAO Yongle.Semi-Implicit FEM for the Valuation of American Options Under the Heston Model[J].Computational and Applied Mathematics,2022,41(2):73.DOI:10.1007/s40314-022-01764-y.
[6] ZHANG Tie,ZHANG Shuhua,ZHU Danmei.Finite Difference Approximation for Pricing the American Lookback Option[J].Journal of Computational Mathematics,2009,27(4):484-494.
[7] 张琪,高景璐.美式回望看涨期权的有限元方法[J].吉林大学学报（理学版）,2014,52(6):1167-1170.
[8] SONG Haiming,ZHANG Qi,ZHANG Ran.A Fast Numerical Method for the Valuation of American Lookback Put Options[J].Communications in Nonlinear Science and Numerical Simulation,2015,27(1-3):302-313.
[9] SONG Haiming,WANG Xiaoshen,ZHANG Kai,et al.Primal-Dual Active Set Method for American Lookback Put Option Pricing[J].East Asian Journal on Applied Mathematics,2017,7(3):603-614.
[10] MOTAMENI M,MEHRDOUST F,NAJAFI A R.Lookback Option Pricing Under the Double Heston Model Using a Deep Learning Algorithm[J].Computational and Applied Mathematics,2022,41(8):378.DOI:10.1007/s40314-022-02098-5.
[11] BENZI R,SUCCI S,VERGASSOLA M.The Lattice Boltzmann Equation:Theory and Applications[J].Physics Reports,1992,222(3):145-197.
[12] 何雅玲,王勇,李庆.格子Boltzmann方法的理论及应用[M].北京:科学出版社,2009:31-55.
[13] CHAI Zhenhua,LIANG Hong,DU Rui,et al.A Lattice Boltzmann Model for Two-Phase Flow in Porous Media[J].SIAM Journal on Scientific Computing,2019,41(4):B746-B772.
[14] ZHAO Yong,PEREIRA GERALD G,KUANG Shibo,et al.A Generalized Lattice Boltzmann Model for Solid-Liquid Phase Change with Variable Density and Thermophysical Properties[J].Applied Mathematics Letters,2020,104:106250.DOI:10.1016/j.aml.2020.106250.
[15] SUN Dongke.A Discrete Kinetic Scheme to Model Anisotropic Liquid-Solid Phase Transitions[J].Applied Mathematics Letters,2020,103:106222.DOI:10.1016/j.aml.2020.106222.
[16] SHI Baochang,DENG Bin,DU Rui,et al.A New Scheme for Source Term in LBGK Model for Convection-Diffusion Equation[J].Computers & Mathematics with Applications,2008,55(7):1568-1575.
[17] 冯颖欣,戴厚平,汪辰.扩展Fisher-Kolmogorov方程的格子Boltzmann方法[J].吉首大学学报（自然科学版）,2023,44(4):19-30.
[18] WU Fangfang,XU Duoduo,WANG Yingying.Lattice Boltzmann Model for a Class of Coupled Nonlinear Partial Differential Equations with Variable Coefficients[J].Physica Scripta,2022,98(1):015219.DOI:10.1088/1402-4896/ac9497.
[19] WU Fangfang,XU Duoduo,TIAN Ming,et al.Lattice Boltzmann Method for the Generalized Black-Scholes Equation[J].Advances in Mathematical Physics,2023,2023(1):1812518.DOI:10.1155/2023/1812518.
[20] GOLDMAN M BARRY,SOSIN HOWARD B,GATTO MARY ANN.Path Dependent Options:"Buy at the Low,Sell at the High"[J].The Journal of Finance,1979,34(5):1111-1127.
[21] WILMOTT P,DEWVNNE J,HOWISON S.Option Pricing:Mathematical Models and Computation[M].Oxford:Oxford Financial Press,1993:1.
[22] ZHANG Ran,ZHANG Qi,SONG Haiming.An Efficient Finite Element Method for Pricing American Multiasset Put Options[J].Communications in Nonlinear Science and Numerical Simulation,2015,29(1/2/3):25-36.
[23] SAPARIUC I,MARCOZZI M D,FLAHERTY J E.A Numerical Analysis of Variational Valuation Techniques for Derivative Securities[J].Applied Mathematics and Computation,2004,159(1):171-198.
[24] QIAN Y H,D'HUMIRES D,LALLEMAND P.Lattice BGK Models for Navier-Stokes Equation[J].Europhysics Letters,1992,17(6):479-484.
[25] STERLING JAMES D,CHEN Shiyi.Stability Analysis of Lattice Boltzmann Methods[J].Journal of Computational Physics,1996,123(1):196-206.
[26] CHAI Zhenhua,SHI Baochang.Multiple-Relaxation-Time Lattice Boltzmann Method for the Navier-Stokes and Nonlinear Convection-Diffusion Equations:Modeling,Analysis,and Elements[J].Physical Review E,2020,102(2):023306.DOI:10.1103/PhysRevE.102.023306.
[27] WOLF-GLADROW D A.Lattice-Gas Cellular Automata and Lattice Boltzmann Models:An Introduction[M]∥WOLF-GLADROW D A.Lecture Notes in Mathematics,Vol.1725.Berlin:Springer,2000:1-314.
[28] GUO Zhaoli,ZHENG Chuguang,SHI Baochang.Non-Equilibrium Extrapolation Method for Velocity and Pressure Boundary Conditions in the Lattice Boltzmann Method[J].Chinese Physics,2002,11(4):366-374.
[29] 张琪,张然,宋海明.美式回望期权定价问题的有限体积法[J].物理学报,2015,64(7):070202.DOI:10.7498/aps.64.070202.
[30] BOYLE P,DRAVIAM T.Pricing Exotic Options Under Regime Switching[J].Insurance:Mathematics and Economics,2007,40(2):267-282.