• least squares
• conditional expectation function
• optimal stopping rule
• Options
• ito's lemma
• Monte Carlo method
• high dimensional put
• stochastic processes
• early exercise value

With regard to a particular derivatives instruments, the famous Black-Scholes model development on 1973 provides an explicit closed form solution for the value of this derivative. However this procedure allow an analytical solution to the problem only when it is applied to a particular type of options defined European. Here plays an important role the time on which the option can be exercised, that for the European option is allowed only at the expiry date. Due to the differences in this characteristic the procedure is not applicable to another style option called American: in this case the exercise is allowed not only at the maturity but at any observable time before the expiry. For this reason the pricing of American option become a problem of optimal exercise of the derivative, and in these cases it is inevitable to occur to numerical techniques to find a result. Monte Carlo simulation have been proposed to address this problem, especially when the underlying dinamics are very complex and the classic methods are not adequate to find a solution, in particular where the value of option depends to multiple sources of uncertainty, that is, when the underlying is represented by two or more assets traded into the market. The purpose of this work, after introducing the main concepts related to options, is to discuss some of the recent applications of Monte Carlo approach applied to American option pricing problems. The main contribution have been developed by Longstaff and Schwartz (2001) that have proposed an algorithm for valuing American put option and the early exercise value. Recalling the idea of optimal stopping problem, they start with simulation of possible paths for the stock price and use a simple least squares regression to estimate the conditional expectation function . Estimating this function for each exercise date, obtain a complete specification of the optimal exercise strategy along each path. Subsequently, American option is valued discounting all optimal strategies at the initial time and making an average of paths