In terms of theory, Monte Carlo valuation relies on risk neutral valuation. An option on equity may be modelled with one source of uncertainty: the price of the underlying stock in question. In other cases, the source of uncertainty may be at a remove. Monte Binary option payoffs Methods allow for a compounding in the uncertainty.
Here, correlation between asset returns is likewise incorporated. Least Square Monte Carlo is used in valuing American options. The technique works in a two step procedure. First, a backward induction process is performed in which a value is recursively assigned to every state at every timestep.
Secondly, when all states are valued for every timestep, the value of the option is calculated by moving through the timesteps and states by making an optimal decision on option exercise at every step on the hand of a price path and the value of the state that would result in. This second step can be done with multiple price paths to add a stochastic effect to the procedure. Although the term ‘Monte Carlo method’ was coined by Stanislaw Ulam in the 1940s, some trace such methods to the 18th century French naturalist Buffon, and a question he asked about the results of dropping a needle randomly on a striped floor or table. Fabozzi: Valuation of fixed income securities and derivatives, pg. Valuing American options by simulation: a simple least squares approach”.