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. An important development was the introduction in 1996 by Carriere of Monte Carlo methods for options 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: Least Square Monte Carlo. Least Square Monte Carlo is a technique for valuing early-exercise options (i.e.  Bermudan or American options). It was first introduced by Jacques Carriere in 1996. It is based on the iteration of a two step procedure: Application. 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. Conversely, however, if an analytical technique for valuing the option exists—or even a numeric technique, such as a (modified) pricing tree—Monte Carlo methods will usually be too slow to be competitive. They are, in a sense, a method of last resort; see further under Monte Carlo methods in finance. With faster computing capability this computational constraint is less of a concern. References. Notes Sources Primary references Bibliography External links. Online tools Discussion papers and documents