The Application Of The Least-Squares Monte Carlo Methode In Pricing American Options

The pricing of American options is always one of the most important and popular problem in the discipline of financial mathematics. In 2001, F. A. Longstaff and E. S. Schwartz proposed a new approach for approximating the value of American options by estimating the conditional expected payoff using Least Squares.This article presents the characters of American Options and of the Least Squares Monte Carlo and it develops the Least Squares Monte Carlo approach by some numerical techniques. Using C++ for programming, some numerical results of two types of American options are given at the end of this article.For the numerical improvements, firstly, with the help of the Brownian Bridge simulating all the paths at one time and saving them during all the procedure of computation are not necessary any more, so we can economize the memory space and accelerate the computation. The stock price can be backward simulated using Brownian Bridge. Because the distribution of the stock price is determined when its initial value and finale value are fixed, so it needs only save the two moments'prices during all the recursive stepsSecondly, with the help of variance reduction method, we can narrow the boundary of the confidence interval for augmenting the estimation's accuracy. The key to the variance reduction is to find out a stochastic variables Y satisfied E ( X ) = E (Y ) + E ( X ? Y)andvar( X ? Y ) < var( X). For the pricing examples in this article, we chose the European option asY . Comparing the two different numerical results the standard error of the method with control variate is much smaller than that without it, from which we can conclude that this method is effective.