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. An important development was the introduction in 1996 by Carriere of Monte Carlo methods for options with early exercise features. Methodology 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: An option on equity may be modelled with one source of uncertainty: the price of the underlying stock in question. Here the price of the underlying instrument is usually modelled such that it follows a geometric Brownian motion with constant drift and volatility . So: , where is found via a random sampling from a normal distribution; see further under Black–Scholes. Since the underlying random process is the same, for enough price paths, the value of a european option here should be the same as under Black–Scholes. More generally though, simulation is employed for path dependent exotic derivatives, such as Asian options. In other cases, the source of uncertainty may be at a remove. For example, for bond options the underlying is a bond, but the source of uncertainty is the annualized interest rate (i.e. the short rate). Here, for each randomly generated yield curve we observe a different resultant bond price on the option's exercise date; this bond price is then the input for the determination of the option's payoff. The same approach is used in valuing swaptions, where the value of the underlying swap is also a function of the evolving interest rate. (Whereas these options are more commonly valued using lattice based models, as above, for path dependent interest rate derivatives – such as CMOs – simulation is the