**3. Discussion and Conclusion**

The FOST can be thought of as generalizations of two existing pricing methodologies. First, it generalizes the Forest of Trees method to a high-dimensional underlying. Second, it generalizes the Stochastic Tree pricing method for single-exercise right options to multiple exercise rights. We construct high and low FOST estimators analogous to those defined for the Stochastic Tree. We prove properties regarding FOST estimator bias, ordering, and convergence and present numerical results as illustrations.

In related work, we have replaced the binomial/trinomial trees in the Forest of Trees method with Stochastic Meshes Broadie and Glasserman (2004), creating the Forest of Stochastic Meshes Marshall and Reesor (2011). This avoids the exponential growth in computing time with the number of exercise opportunities experienced by the FOST. Another avenue of future work involves algorithmic enhancement. In Section 2.5 we discussed the use of parallel processing to reduce computing time. Two alternatives to this are variance reduction and bias reduction. There are some standard variance reduction methods (e.g., antithetic variates, control variates) that could be used to produce more efficient estimators. The bias reduction technique given in Whitehead et al. (2012) for Stochastic Tree estimators successfully reduces the branching factor required to obtain a desired accuracy for an American option value. This technique can be extended to correct the bias in FOST estimators and we have preliminary evidence of its effectiveness Marshall (2012). Combinations of variance reduction, bias reduction, and parallel processing can be investigated to further improve the algorithm's performance.

**Author Contributions:** R.M.R. and T.J.M. contributed equally to all aspects of this paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded in part by the Natural Sciences and Engineering Research Council of Canada gran<sup>t</sup> number RGPIN-2017-05441.

**Acknowledgments:** The authors thank SHARCNet for computational resources and technical support, particular acknowledgement goes to both Tyson Whitehead and Baolai Ge. We also thank graduate students and faculty in the Financial Mathematics group at Western University, in particular, Matt Davison, Adam Metzler, Rogemar Mamon, and Lars Stentoft.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. The views represented herein are the authors' own views and do not necessarily represent the views of Bank of Montreal or its affiliates and are not a product of its research.
