Monte Carlo methods for option pricing

In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features. The first application to option pricing was by Phelim Boyle in 1977 (for European options). In 1996, M. Broadie and P. Glasserman showed how to price Asian options by Monte Carlo. In 2001 F. A. Longstaff and E. S. Schwartz developed a practical Monte Carlo method for pricing American-style options.

In terms of theory, Monte Carlo valuation relies on risk neutral valuation. Here the price of the option is its discounted expected value; see risk neutrality and rational pricing. The technique applied then, is (1) to generate a large number of possible, but random, price paths for the underlying (or underlyings) via simulation, and (2) to then calculate the associated exercise value (i.e. "payoff") of the option for each path. (3) These payoffs are then averaged and (4) discounted to today. This result is the value of the option.

This approach, although relatively straightforward, allows for increasing complexity:

Least Square Monte Carlo is used in valuing American options. The technique works in a two step procedure.

As can be seen, Monte Carlo Methods are particularly useful in the valuation of options with multiple sources of uncertainty or with complicated features, which would make them difficult to value through a straightforward Black–Scholes-style or lattice based computation. The technique is thus widely used in valuing path dependent structures like lookback- and Asian options and in real options analysis. Additionally, as above, the modeller is not limited as to the probability distribution assumed.

...
Wikipedia