Pricing a Bermudan Option with the Longstaff-Schwartz Monte Carlo Method
A Bermudan put option on a stock gives its holder the right to sell the stock at an agreed strike price at a certain finite number of fixed times before or at the final expiry time. Thus a Bermudan put option is more valuable than a European option (with the same parameters) but less valuable than an American put option, which can be exercised at any time before expiry. This Demonstration implements the famous method due to Longstaff and Schwartz of computing the price of a Bermudan put option on a stock by Monte Carlo simulation. Although the method can be applied to any model of stock movement, here we use it in the case of the classical Black–Scholes model. For simplicity, we also assume that the stock pays no dividend.
In this Demonstration the option can be exercised at any of three moments in time prior to the expiry (at time 1). The time of exercise can be changed by moving the three colored points along the time axis. You can vary the strike price of the option (represented by the black horizontal line), as well as the initial stock price, its volatility, and the rate of appreciation (assumed, by the principle of risk neutrality to be equal to the rate of interest). The red points on some of the paths, directly above the exercise points on the time axis, correspond to stock values at which it is optimal to exercise the option at that particular exercise time.
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.