*2.2. Estimator Bias*

In order to justify using the high- and low-biased estimators to construct upper and lower option price bounds, respectively, we prove that the high estimator is always positively biased and that the low estimator is always negatively biased. In addition we include a comparison of the estimators which orders their values on any realization of the simulated forest.

The theorems that follow are direct extensions of those in Broadie and Glasserman (1997). Below, the branching factor, *b*, appears as an argumen<sup>t</sup> in the estimators. For example, *<sup>V</sup>*ˆ0(*b*, **S**0, N0, *<sup>U</sup>*0) refers to the time-0, state-Z0 high-biased estimator with a stochastic tree branching factor of *b*. This argumen<sup>t</sup> has been suppressed to this point for convenience. We begin with the theorem regarding the bias of the high estimator.

**Theorem 1.** *(High estimator bias) The high estimator is biased high, i.e.,*

$$\mathbb{E}\left[\hat{V}\_0\left(b,\mathbf{S}\_{0\prime}\mathcal{N}\_{0\prime}\mathcal{U}\_0\right)\right] \geq B\_0\left(\mathbf{S}\_{0\prime}\mathcal{N}\_{0\prime}\mathcal{U}\_0\right) \tag{13}$$

*for all b.*

> Similarly, the result stating the bias of the low estimator follows.

**Theorem 2.** *(Low estimator bias) The low estimator is biased low, i.e.,*

$$\mathbb{E}\left[\vartheta\_{0}\left(b,\mathbf{S}\_{0\prime},\mathcal{N}\_{0\prime}\mathcal{U}\_{0\prime}\right)\right] \leq B\_{0}\left(\mathbf{S}\_{0\prime},\mathcal{N}\_{0\prime}\mathcal{U}\_{0}\right) \tag{14}$$

*for all b.*

> Finally, an ordering result for the high and low estimators is stated in Theorem 3.

**Theorem 3.** *(Comparison of Estimators) On every realization of the forest the low estimator is less than or equal to the high estimator. That is,*

$$\left(\boldsymbol{\psi}\_{i}\left(\boldsymbol{b}\_{\prime}\mathbf{S}\_{i\prime\prime}^{\mathbf{j}}\mathcal{N}\_{i\prime}\mathcal{U}\_{i}\right)\right)\leq\mathcal{V}\_{i}\left(\boldsymbol{b}\_{\prime}\mathbf{S}\_{i\prime\prime}^{\mathbf{j}}\mathcal{N}\_{i\prime}\mathcal{U}\_{i}\right)\tag{15}$$

*with probability one for all* **j***, i.*

> Theorems 1–3 are proven in Appendix B.
