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For a long time it used to be believed that the Monte Carlo method is not suitable for pricing the American type of option (one can still find this claim repeated in older texts on mathematical finance). The principle lay in efficiently estimating conditional expectations. The crucial idea, of using least-squares regression on a finite set of functions, was implemented by F. A. Longstaff and E. S. Schwartz (having been earlier proposed by J. Carrière). In this Demonstration we implement the Longstaff and Schwartz algorithm for the standard Bermudan put and call options in the Black–Scholes model. An American option can be treated as a limit of Bermudan options, so by computing the value of a Bermudan option with a large number of exercise times one can obtain a good approximation to the value of the American option.

The number of sample paths used in this Demonstration is too small to expect an accurate value. As a rough measure of accuracy obtained from a particular sample, the value of the corresponding European option, computed by means of the Black–Scholes formula, is displayed below the estimated value of the Bermudan option.

Note that if we place all the exercise times at the expiry time, the Bermudan option will turn into a standard European one, whose value is given by the Black–Scholes formula. This suggests the following method of pricing Bermudan options, which ought to produce reasonably accurate answers even with a small number of sample paths. First place all the exercise points at the expiry time and generate successive samples until the estimate is close to the value given by the Black–Scholes formula. Once that happens, move the exercise times to desired locations.

J. Carrière, "Valuation of Early-Exercise Price of Options Using Simulations and Nonparametric Regression," Insurance: Math. Econ., 19, pp. 19–30 1988.

F. A. Longstaff and E. S. Schwartz, "Valuing American Options by Simulation: A Simple Least-Squares Approach," Review of Financial Studies, 14(1), pp. 113–147 2001.