**1. Introduction**

The broad class of stochastic optimal control problems includes many important applications in managemen<sup>t</sup> sciences and quantitative finance such as development of natural resources, project initiation or abandonment, maintenance scheduling, land use decisions, and valuation and hedging of complex contracts. Very few problems have closed form solutions. For example the Black–Scholes–Merton formula for pricing European-style options does not have an American-style option analog. Approaches such as binomial lattice methods, partial differential equation (PDE) methods, variational inequalities and integral equations have been adopted for pricing these types of derivatives. However, all of these methods mentioned are limited in the number of sources of uncertainty and the dimensionality of the underlying asset that can be practically incorporated, leaving Monte Carlo (MC) as a general method of solution.

Multiple exercise options (MEOs) are generalizations of American-style options as they provide the holder more than one exercise right and sometimes control over one or more other variables, such as the amount exercised. Similar to pricing American-style options, MEO valuation is a problem in stochastic optimal control. For American-style options, the solution provides both a value and optimal exercise rule—a stopping time. For MEOs the solution gives both a value and optimal exercise policy. In cases in which the holder controls only the exercise times, the exercise policy is a sequence of stopping times. For MEOs in which the holder controls the exercise times and amounts, the exercise policy is a paired sequence of stopping times and exercise amounts. The policy generalizes to other control variables. Multiple exercise option valuation algorithms are generalizations of those used for pricing American-style options. Most of the work in the literature has focused on swing options in which the holder controls the exercise times and amounts. Hence, we discuss MEO valuation in the

context of swing options and note that the FOST method and the FOST price estimators' properties apply to general MEOs.

The Forest of (Binomial/Trinomial) Trees method of Lari et al. (2001) and Jaillet et al. (2004) extends the binomial method of Cox et al. (1979) for pricing American style options to value swing options. As with most tree-based methods, the Forest of Trees is not able to handle high-dimensional underlying processes. Tangential to the Forest of Trees extension, the Stochastic Tree method of Broadie and Glasserman (1997) extends the binomial method for pricing American style options to allow for a high-dimensional underlying asset.

In this paper, we replace the binomial and trinomial trees of Lari et al. (2001) and Jaillet et al. (2004) with the stochastic trees of Broadie and Glasserman (1997), hence creating the Forest of Stochastic Trees (FOST) method for valuing multiple exercise options. The FOST can be thought of as generalizations of two different methodologies; specifically, it extends


Visually these generalizations complete the final entry in the two-by-two table shown in Table 1.


**Table 1.** Table showing that the Forest of Stochastic Trees extends (i) the Forest of Trees method to a high-dimensional underlying; and (ii) the Stochastic Tree method to multiple exercise options.

We construct high and low biased FOST estimators that are analogous to those from the Stochastic Tree. The main theoretical results, presented in Sections 2.2 and 2.3 and proven Appendix B, are listed below.


The remainder of this section provides a literature review. Section 2 describes the Forest of Stochastic Trees estimators in detail; specifies results giving estimator properties (e.g., biasedness and convergence); provides numerical results showing the method is effective at pricing; and supplies discussion on improving computational performance through parallel processing. Section 3 summarizes and concludes the paper. A list of the notation used is given in Appendix A and proofs of the paper's main theoretical contributions on estimator properties are given in Appendix B.
