The Least-Square Monte Carlo Method Of American Options Pricing And Its Improved Models

This article describes the theoretical characteristics of the American option and the mechanism of the least squares Monte Carlo method, then studies the improvement of numerical methods based on least squares Monte Carlo method, obtained four improved model, and finally uses MATLAB programming to get the numerical results of several improved model for pricing American options.First, due to the Monte Carlo method has a large fluctuation variance, which will affect the accuracy of the simulation, this paper presents the Halton sequence, Faure sequence and Sobol sequence algorithm. And then the Sobol sequence based on the low bias sequence is used in place of the pseudo-random number in the Monte Carlo method, and obtains the least squares quasi-Monte Carlo method (LSQM).Secondly, we have improved the quasi-Monte Carlo method from two current development directions:One is the randomized quasi-Monte Carlo method, means to randomize the deterministic Sobol low discrepancy sequence, so that can obtain sequence of random numbers which is both low discrepancy and stochastic. The other is the effective dimension reduction method. We mainly discuss the brownian bridge and principle components analysis, then synthesis these two technologies to get three improved models: LSQM based on brownian bridge (LSQM_BB), LSQM based on principle components analysis (LSQM_PCA), and a hybrid LSQM based on brownian bridge and principle components analysis (LSQM_BB&PCA).Finally, with the help of MATLAB program, we have empirical analysis for these models in terms of option pricing, standard deviation and run time, and draw relevant conclusions.