**2. Results**

We consider the valuation of multiple exercise options as a stochastic optimal control problem with three relevant state variables—the underlying variable (*S*), number of exercise rights remaining ( N ), and usage level ( *U*) assuming some volume control. At each exercise opportunity and given (*S*, N , *U*), the current values of the state variables, the holder must choose between


Note that with volume control the payoff from exercising *u* units changes with *u* (as does the continuation value of the option). Thus, the holder chooses the value-maximizing *u* when deciding to exercise. Also note that with N = 1 and *u* constrained to be 1, this is an American-style option.

We work with the time-discretized problem and use dynamic programming to solve for the optimal exercise policy and the corresponding optimal value. In all variables, let the subscript *i* denote time-*ti* and let U*i* be the time-*ti* set of admissible volume choices which includes the zero volume choice (i.e., hold). The recursive equations for the dynamic program are

$$H\_i(\mathcal{S}\_i, \mathcal{N}\_{i+1}, \mathcal{U}\_{i+1}) = \mathcal{E}[B\_{i+1}(\mathcal{S}\_{i+1}, \mathcal{N}\_{i+1}, \mathcal{U}\_{i+1}) | \mathcal{Z}\_i] \qquad \text{and} \tag{1}$$

$$B\_i(\mathbb{S}\_{i\prime}\mathcal{N}\_i, \mathcal{U}\_i) = \max\_{u \in \mathcal{U}\_i} \left[ h\_i(\mathbb{S}\_{i\prime}\mathcal{N}\_i, \mathcal{U}\_{i\prime}u) + H\_i(\mathbb{S}\_{i\prime}\mathcal{N}\_i - I\_{\{u \neq 0\}\prime}, \mathcal{U}\_i + u) \right],\tag{2}$$

with the terminal conditions

$$H\_m(S\_m, \mathcal{N}\_m, \mathcal{U}\_m) = \tilde{\Phi} \left( \mathcal{U}\_m \right) \qquad \text{and} \tag{3}$$

$$B\_{\mathfrak{m}}(\mathbb{S}\_{\mathfrak{m}\prime}\mathcal{N}\_{\mathfrak{m}\prime}\mathcal{U}\_{\mathfrak{m}\prime}) = \max\_{\mathfrak{u}\in\mathcal{U}\_{\mathfrak{m}}} \left[ h\_{\mathfrak{m}}(\mathbb{S}\_{\mathfrak{m}\prime}\mathcal{N}\_{\mathfrak{m}\prime}\mathcal{U}\_{\mathfrak{m}\prime}\boldsymbol{\mu}) + H\_{\mathfrak{m}}(\mathbb{S}\_{\mathfrak{m}\prime}\mathcal{N}\_{\mathfrak{m}\prime} - \mathbb{I}\_{\{\mathfrak{u}\neq 0\}}, \mathcal{U}\_{\mathfrak{m}} + \boldsymbol{\mu}) \right],\tag{4}$$

where *Hi*(*<sup>S</sup>*, N , *U*) and *Bi*(*<sup>S</sup>*, N , *U*) are the time-*ti*, state-Z*i* continuation and option values, respectively, *hi*(*<sup>S</sup>*, N , *U*, *u*) is the payoff from exercising *u* units with *hi*(*<sup>S</sup>*, N , *U*, 0) = 0, Z*i* is the time-*ti* information set generated by the paths of (*S*, N , *U*), *I* is an indicator function and *φ*˜ (·) is a cumulative usage penalty term. Estimator properties and their proofs are given for this multiple exercise option setup. However, the dynamic program and estimator properties can be stated and proven for alternative specifications provided there is a finite number of exercise rights and usage levels. For example, a swing option contract may specify a certain number of *up* and *down* swing rights, N*u* and N*<sup>d</sup>*. An up swing right allows the holder to take more than the baseline amount of the underlying asset while a down swing right allows the holder to take less. Another variation is to allow for multiple rights to be exercised at each opportunity where each right corresponds to a fixed volume amount Bender and Schoenmakers (2006); Meinshausen and Hambly (2004).

### *2.1. Forest of Stochastic Trees*

The FOST generalizes the stochastic tree method for valuing American-style options to the valuation of multiple exercise options and extends the Forest of Trees method to handle a high-dimensional underlying asset. This is done by replacing the binomial/trinomial trees with stochastic trees in the framework of Lari et al. (2001) and Jaillet et al. (2004) hence giving the FOST. The stochastic tree is constructed identically as described in Broadie and Glasserman (1997) and the tree is replicated multiple times, with one replication corresponding to each possible ( N , *U*) combination. This is analogous to the Forest of Trees in which the same underlying binomial/trinomial tree is replicated for each possible ( N , *U*) combination.

The dynamic program is approximately solved by replacing the continuation values in Equations (1) and (3) with stochastic tree-type estimators. As with the original stochastic tree technique, high- and low-biased option value estimators are constructed by using the analogous high- and low-biased estimators, respectively, on each stochastic tree in the forest. The recursive equations for the high estimator are

$$
\hat{H}\_i(\mathbf{S}\_{i'}^\dagger \mathcal{N}\_{i+1\prime} \mathcal{U}\_{i+1}) = \frac{1}{b} \sum\_{k=1}^b \hat{\mathcal{V}}\_{i+1}(\mathbf{S}\_{i+1\prime}^\mathbf{k} \mathcal{N}\_{t+1\prime} \mathcal{U}\_{t+1})\_\prime \quad \text{and} \tag{5}
$$

$$\hat{V}\_{i}(\mathbf{S}\_{i}^{\mathbf{j}}, \mathcal{N}\_{i}, \mathcal{U}\_{i}) = \max\_{u \in \mathcal{U}\_{i}} \left[ h\_{i}(\mathbf{S}\_{i}^{\mathbf{j}}, \mathcal{N}\_{i}, \mathcal{U}\_{i}, u) + \hat{H}\_{i}(\mathbf{S}\_{i}^{\mathbf{j}}, \mathcal{N}\_{i} - I\_{\{u \neq 0\}}, \mathcal{U}\_{i} + u) \right],\tag{6}$$

with the terminal conditions

$$\mathcal{V}\_{m}(\mathbf{S}\_{m\prime}^{\dagger}, \mathcal{N}\_{m\prime} \cup\_{m} \mathcal{U}\_{m}) = \max\_{u \in \mathcal{U}\_{m}} \left[ h\_{m}(\mathbf{S}\_{m\prime}^{\dagger}, \mathcal{N}\_{m\prime} \cup \mathcal{U}\_{m\prime}, u) + \tilde{\phi} \left( \mathcal{U}\_{m} + u \right) \right], \tag{7}$$

where *<sup>H</sup>*<sup>ˆ</sup>*i*(**<sup>S</sup>**, N , *U*) and *V*ˆ *<sup>i</sup>*(**<sup>S</sup>**, N , *U*) are the time-*ti*, state-Z*i* continuation and option value estimators, respectively, *hi*(**<sup>S</sup>**, N , *U*, *u*) (with *hi*(**<sup>S</sup>**, N , *U*, 0) = 0) is the time-*ti*, state-Z*i* payoff from exercising *u* units, *b* is the branching factor, *I* is an indicator function and *φ*˜ (*Um* + *u*) is a global usage penalty term. The superscript **j** = {*j*0, *j*1, ... , *ji*} indicates the specific node within a given stochastic tree and **k** = {**j**, *k*}.

Figure 1 is a diagram of a section of a Forest of Stochastic Trees with two volume choices, *u*1 and *u*2. It illustrates the nodes in the forest which need to be considered when making an exercise decision given state ( N , *U*). The three choices are no exercise, exercise *u*1 units, and exercise *u*2 units.

**Figure 1.** Section of a Forest of Trees with N = # of exercise rights remaining, *U* = usage level, and three exercise choices—no exercise, excerise *u*1 units, and exercise *u*2 units.

The low estimator is similarly defined using the low estimator on each stochastic tree via the dynamic program,

$$\mathfrak{F}\_{\rm il}(\mathbf{S}\_{\rm i}^{\rm j}, \mathcal{N}\_{\rm i}, \mathcal{U}\_{\rm i}, \boldsymbol{\mu}) = h\_{\rm i} \left( \mathbf{S}\_{\rm i}^{\rm j}, \mathcal{N}\_{\rm i}, \mathcal{U}\_{\rm i}, \boldsymbol{\mu} \right) + \frac{1}{b - 1} \sum\_{\substack{k = 1 \\ k \neq \boldsymbol{1}}}^{b} \mathfrak{E}\_{\rm i + 1} (\mathbf{S}\_{\rm i + 1}^{\rm k}, \mathcal{N}\_{\rm i} - \mathcal{I}\_{\{\boldsymbol{u} \neq \boldsymbol{0}\}}, \mathcal{U}\_{\rm i} + \boldsymbol{\mu}), \tag{8}$$

$$\hat{H}\_{\text{il}}(\mathbf{S}\_{i'}^{\mathbf{j}}, \mathcal{N}\_{i}, \mathcal{U}\_{i}) = \max\_{u \in \mathcal{U}\_{i}} \left[ \xi\_{\text{il}}(\mathbf{S}\_{i'}^{\mathbf{j}}, \mathcal{N}\_{i} - I\_{\{u \neq 0\}}, \mathcal{U}\_{i} + u) \right],\tag{9}$$

$$\mathfrak{d}\_{i\bar{l}}(\mathbf{S}\_{\mathbf{i}'}^{\bar{\mathbf{j}}}, \mathcal{N}\_{\mathbf{i}'} \mathcal{U}\_{\mathbf{i}}) = h\_{\mathbf{i}}(\mathbf{S}\_{\mathbf{i}'}^{\bar{\mathbf{j}}}, \mathcal{N}\_{\mathbf{i}'} \mathcal{U}\_{\mathbf{i}'} \mathcal{U}^\*) + \mathfrak{d}\_{\mathbf{i}+1}(\mathbf{S}\_{\mathbf{i}+1'}^{\mathbf{1}}, \mathcal{N}\_{\mathbf{i}} - \mathcal{I}\_{\{\mathbf{u}^\* \neq 0\}'} \mathcal{U}\_{\mathbf{i}} + \mathcal{U}^\*), \quad \text{and} \tag{10}$$

$$\mathfrak{d}\_{i}\mathfrak{d}\_{i}(\mathbf{S}\_{i}^{\dagger},\mathcal{N}\_{i},\mathcal{U}\_{i}) = \frac{1}{b} \sum\_{l=1}^{b} \mathfrak{d}\_{il}(\mathbf{S}\_{i}^{\dagger},\mathcal{N}\_{l},\mathcal{U}\_{l}) \tag{11}$$

where *H* ˆ *il*(**X***ji*, N*<sup>i</sup>*, *Ui*) is the *l*−th leave-one-out hold value estimator and *u*<sup>ˆ</sup><sup>∗</sup> is the estimated optimal exercise amount which depends on *i* and *l*. The terminal conditions associated with this dynamic programming scheme are,

$$\mathfrak{d}\_{\mathfrak{m}}(\mathbf{S}\_{\mathfrak{m}}^{\mathbf{j}}, \mathcal{N}\_{\mathfrak{m}}, \mathcal{U}\_{\mathfrak{m}}) = \max\_{\mathfrak{u} \in \mathcal{U}\_{\mathfrak{U}}} \left[ h\_{\mathfrak{m}}(\mathbf{S}\_{\mathfrak{m}}^{\mathbf{j}}, \mathcal{N}\_{\mathfrak{m}}, \mathcal{U}\_{\mathfrak{m}}, \mathfrak{u}) + \tilde{\mathfrak{g}} \left( \mathcal{U}\_{\mathfrak{m}} + \mathfrak{u} \right) \right], \tag{12}$$

where *φ*˜ (*Um* + *u*) is a cumulative usage penalty term.
