The Application Of The LSM Method Combined With QMC Method In Pricing American Options

As the most frequently used financial derivatives,options play an irreplaceable role in risk management,hedging,speculation and arbitrage.In the option transaction,because the price of the option will directly affect the profit and loss of the buyer and seller,how to use the mathematical model to determine the price of the option is the most important core issue.The Black-Scholes-Merton model proposed in 1973 is a milestone in the history of financial mathematics.It gives an explicit solution to the pricing problem of standard European options.However,it is very difficult to find analytical solutions for American options,which is more actively traded in the market,so in most cases we can only rely on numerical methods to study the pricing of American options.And because the characteristics of American options can be executed in advance,how to determine the optimal exercise time is the key to determine the price.At present,there are mainly three numerical methods for the pricing of American op-tions:the binary-tree method,the finite difference method and the Monte Carlo method.The first two methods may suffer from "Curse of dimensionality" when dealing with high-dimensional problems,while Monte Carlo method is not affected.The standard way for princing American option using Monte Carlo method is the least-square Monte Carlo method,which uses the simple least-squares approach to determine the return and optimal stopping time simutaneously.But,when dealing with a series of problems,in--cluding high-dimensional problem,the standard Monte-Carlo method does not perform well,and need to be improved in both computational accuracy and efficiency,so many variance reduction techniques have been proposed,as to enhance and optimize the orig-inal method.This thesis will first introduce the least-square Monte Carlo method,and various variance reduction techniques,including Brownian Bridge method and Principal Component Analysis method,and Quasi Monte Carlo method.Then we will combine the LSM method with QMC method,to improve the original method.The result shows that the LSM-QMC method is very effective.