Monte Carlo method really only estimates the option value assuming a given starting point and time. However, for early exercise, we would also need to know the option value at the intermediate times between the simulation start time and the option expiry time. In the Black–Scholes PDE approach these prices are easily obtained, because the simulation runs backwards from the expiry date. In Monte-Carlo this information is harder to obtain, but it can be done for example using the least squares algorithm of Carriere (see link to original paper) which was made popular a few years later by Longstaff and Schwartz (see link to original paper). Monte Carlo methods Mathematically The fundamental theorem of arbitrage-free pricing states that the value of a derivative is equal to the discounted expected value of the derivative payoff where the expectation is taken under the risk-neutral measure [1]. An expectation is, in the language of pure mathematics, simply an integral with respect to the measure. Monte Carlo methods are ideally suited to evaluating difficult integrals (see also Monte Carlo method). Thus if we suppose that our risk-neutral probability space is and that we have a derivative H that depends on a set of underlying instruments . Then given a sample from the probability space the value of the derivative is . Today's value of the derivative is found by taking the expectation over all possible samples and discounting at the risk-free rate. I.e. the derivative has value: where is the discount factor corresponding to the risk-free rate to the final maturity date T years into the future. Now suppose the integral is hard to compute. We can approximate the integral by generating sample paths and then taking an average. Suppose we generate N samples then which is much easier to compute. Sample paths for standard models In finance, underlying random variables (such as an underlying stock price) are usually assumed to follow a path that is a function of a Brownian motion 2. For example, in the standard Black–Scholes model, the stock price evolves as To sample a path following this distribution from time 0 to T, we chop the time interval into M units of length , and approximate the Brownian motion over the interval by a single normal variable of mean 0 and variance . This leads to a sample path of for each k between 1 and M. Here each is