*Literature Review*

Multiple exercise options arise in many different areas and the structure of these contracts is typically tailored to particular clients/needs, in contrast to standardized derivatives such as interest rate swaps and exchange-traded commodity futures. A non-exhaustive list of examples of MEOs include (i) tolling agreements used in the steel Kim et al. (2019) and electricity Deng and Oren (2006) sectors; (ii) chooser flexible caps which are exotic interest rate derivatives Meinshausen and Hambly (2004); (iii) valuation and control of energy production and storage facilities Chen and Forsyth (2007); Ludkovski and Carmona (2010); Thompson et al. (2009); and (iv) swing options Calvo-Garrido et al. (2017); Jaillet et al. (2004); Lari et al. (2001); Wilhelm and Winter (2008).

Valuation methods for MEOs are extensions of those used for American-style options. There are continuous-time solutions to both the American-style and multiple exercise option valuation problems; these are computed by solving a system of Hamilton–Jacobi–Bellman quasi-variational inequalities Korn et al. (2005). These methods give more accurate and stable price and sensitivity estimates than those computed using simpler tools (e.g., trees). However, these methods are quite complex mathematically and break down in higher dimensions.

In this article, we focus on the mathematically simpler time-discretized version of the valuation problem. Discrete-time tree-based methods for valuing American-style options Cox et al. (1979) have been extended to MEOs via the Forest of Trees Jaillet et al. (2004); Lari et al. (2001). Techniques for pricing American-style options using solutions of PDEs have been modified to MEOs Calvo-Garrido et al. (2017); Chen and Forsyth (2007); Thompson et al. (2009); Wilhelm and Winter (2008). These methods for MEOs inherit properties similar to the corresponding methods for single-exercise options. One crucial property is that these methods fail as the dimensionality of the problem increases.

Monte Carlo is the obvious tool to overcome the curse of dimensionality, as the rate of convergence of Monte Carlo estimators is independent of the dimension. Tilley (1993) was the first to show that the forward-in-time Monte Carlo approach could be used to solve the backward-in-time dynamic programming problem arising from valuation of an American-style option. Since this seminal paper, numerous other methods for the Monte Carlo valuation of American style options have appeared. These include methods that attempt to parameterize the exercise region Barraquand and Martineau (1995) and those that discretize the state space Bally et al. (2005). Methods that parameterize the early-exercise region have been extended to value multiple exercise options by parameterizing the set of exercise level curves Ibánez ¨ (1996). Similarly state space aggregation methods have been used for multiple exercise option valuation Ben Latifa et al. (2016). These approaches, however, also suffer from the curse of dimensionality and do not easily generalize to arbitrary payoffs and underlying price processes.

Monte Carlo methods that do not break down with the dimensionality and that accommodate general payoff and price processes include those that solve the optimal stopping-time problem through estimation of the hold or continuation value. These include the stochastic tree and mesh techniques of Broadie and Glasserman (1997 2004) and the regression-based approach first appearing in Carriere (1996) and then subsequently generalized in Longstaff and Schwartz (2001). For each of these valuation techniques, high- and low-biased estimators are easily generated, along with a hybrid interleaving estimator that has properties of both. Duality-based methods solve the optimal control problem in the dual space by approximating an optimal martingale, typically by regression Andersen and Broadie (2004); Haugh and Kogan (2004).

Least-squares Monte Carlo has been modified for the pricing of swing options in Barrera et al. (2006); Meinshausen and Hambly (2004), respectively. Although increased dimensionality does not decrease the performance of these methods, they suffer from other drawbacks. In least-squares Monte Carlo methods one must select a set of basis functions on which to run regressions to estimate continuation values. In general only a complete (infinite) set of basis functions results in continuation value estimators that are consistent for the true option value. In practice, of course, a finite set of basis functions is used and introduces an approximation error. Continuation value estimators are consistent for the true approximation value and not the true option value Clement et al. (2001); Stentoft (2004).

Duality methods have been extended to MEOs Bender (2011); Chandramouli and Haugh (2012); Gyurko et al. (2015); Meinshausen and Hambly (2004). Duality methods rely on having a sub-optimal exercise policy that produces a low-biased estimate from which the solution to the dual problem can be approximated to yield a high-biased estimate. Typically regression-based methods are used to estimate

the sub-optimal exercise policy Chandramouli and Haugh (2012); Gyurko et al. (2015); Meinshausen and Hambly (2004) implying the above noted issues of least-squares Monte Carlo persist when pricing MEOs. Policy iteration methods such as Bender (2011), yield approximations of the time-0 value at each iteration of the dynamic program. As with the pricing of American-style options this method is advantageous because it removes the requirement to calculate nested conditional expectations prior to the time-0 value being approximated.

The stochastic mesh of Broadie and Glasserman (2004) has been extended to MEOs via the Forest of Stochastic Meshes (FOSM) Marshall (2012); Marshall and Reesor (2011). High and low biased FOSM estimators are derived Marshall and Reesor (2011) and their properties shown Marshall (2012), similar to the work presented here for the Forest of Stochastic Trees estimators.
