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Bermudan and American Style Basket Options Valuation Methods

There are many methods for valuing Bermudan and American style options. The most popular ones are the Rubinstein binomial tree method and its extended version, the Hull-White trinomial tree method. Unfortunately, both methods and other lattice based numerical methods have their limitations. Options that involve multiple factors cannot be valued with these methods, due to the exponential growth of the number of nodes or grids that are used to approximate option values. A lot of effort in the research community of financial engineering has been made to find appropriate methods to value options involving multiple factors. Among the methods that have been researched so far, the most practically useful and popular one is the Monte-Carlo method introduced by Professors Longstaff and Schwartz.

Monte Carlo simulation methods have been widely used in financial instrument valuation. However, their use in option valuation has been mostly restricted to the European style. To value a European style option with Monte Carlo simulation, paths are generated first and intrinsic option values are calculated path by path at the option's maturity date. The option value is then simply the discounted average of the intrinsic values of the paths. Such a simple method is not transferable to a Bermudan or American option. Valuation of such an option requires dynamically determining whether it is optimal to exercise at an exercise date. This cannot be done on a single path. To make an optimal decision, all the paths of the underlyings must be taken into consideration to determine the continuation value of an option and then determine if it is optimal to exercise.

In the Longstaff and Schwartz Monte Carlo method (LSMC), joint paths are first generated at all exercise time points for the assets in the basket. Then one starts from the last exercise date and goes backward step by step to determine the current intrinsic value and the continuation value and thus determining if it is optimal to exercise, given that it is not exercised at the prior time points. For the last exercise date, the optimal exercise value for each path is simply the option's intrinsic value at the date. At other exercise time points, the intrinsic value is also calculated for each joint path. The key is to find the continuation value at an exercise date. According to Longstaff and Schwartz, this can be calculated with the relevant values, denoted Y values, of all joint paths by regressing these values against some properly chosen variables, denoted X variables. The Y values are determined as follows: if a joint path has a future optimal exercise value, the Y value is calculated by discounting the optimal exercise value to the current exercise time point; otherwise, the Y value is simply the intrinsic value at the current exercise time point. The selection of the X variables depends on the model used and also on the option definition as well. For example, for a vanilla call/put option on assets modeled with the Black-Scholes lognormal model, the spot asset prices can be used as the X variables. Another freedom in LSMC is the selection of the regression function, also known as a basis function. Longstaff and Schwartz suggest second order polynomials or second order Laguerre polynomials.

After the regression calculation, the continuation value of the option, i.e., the estimated Y value, is determined for each joint path. By comparing this continuation value with the intrinsic value, one can decide path by path if it is optimal to exercise. If it is, mark this time point as an optimal exercise point and at the same time delete the future optimal mark if there is one. When these steps are finished, calculate the discounted intrinsic value at each optimal exercise time point for each joint path. If there is no optimal exercise time point for a particular path, the option value of the path is 0. The average of the discounted optimal exercise values is then the value of the option.

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