Lattice Boltzmann Method For Pricing American Options

Lattice Boltzmann method is established based on the molecular dynamics theory and is different from conventional numerical methods,which has a clear physical background with complete time,spatial independence,and tractable boundaries,lattice Boltzmann method is a completely parallel mecroscopic simulation algorithm.Lattice Boltzmann method overcomes the shortcomings of many traditional algorithms in solving the numerical solutions of various nonlinear partial differential equations,which has attracted wide attention from many experts and scholars,and the development prospect is increasingly broad owing to its unique advantages.Options,as a core tool for risk management in the financial field,in virtue of the linear complementary model describing American options pricing problem is unbounded and high nonlinear,it is difficult to solve directly.Therefore,most numerical methods are used to solve it.This paper mainly studies applying lattice Boltzmann method to solve the problem of pricing the American options.Firstly,the origin classification of options,the development of options pricing theory and several models describing American options pricing are detailed.Then,the development process of lattice Boltzmann method and its research in various disciplines,and derive the process of establishing lattice Boltzmann model and restoring the Navier-Stokes equation are described.Finally,two improved lattice Boltzmann methods are introduced.Secondly,the linear complementary models that describing standard American put options pricing problem and American multi-asset put options pricing problem are transformed into a parabolic problem using the penalty function method.The unbounded regions of standard American put options pricing problem are transformed into bounded regions by using direct truncation method.Lattice Boltzmann model is established for the transformed equations,and Chapman-Enskog expansion technology is used to select the appropriate equilibrium state distribution function and auxiliary distribution function.D1Q3 model and D1Q5 model are adopted for lattice Boltzmann models to recover the macroscopic equations with second order accuracy and third order accuracy,and the stability analysis of the established lattice Boltzmann models is performed.The unbounded regions of American multi-asset put options pricing problem are transformed into bounded regions by using penalty function method.Lattice Boltzmann model is established for the transformed equations,and D2Q5 model and D2Q9 model are adopted for lattice Boltzmann models to recover the macroscopic equations with second order accuracy and third order accuracy.Finally,the numerical simulation is used to verify the effectiveness of the established model.The numerical simulation results show that option prices and optimal exercise boundary obtained by lattice Boltzmann model and the others existing numerical methods agree well,which verify the effectiveness and practicability of the established model.Lattice Boltzmann model can also be generalized to price more different options.