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There is a rich history of pricing American options by simulation in the literature, beginning with Tilley [21] that first dispelled the belief that simulation could not be used for their valua- tion. Much subsequent work was inspired by this paper. Many methods estimate the continua- tion value of the contingent claim to solve the optimal-stopping-time problem arising from this application. These include Broadie and Glasserman’s stochastic tree [5] and stochastic mesh [6], and the least-squares Monte Carlo (LSM) method in Longstaff and Schwartz [16]. The LSM method is very popular in practice since it is easy to implement and represents nothing more than a cross-sectional regression to estimate the continuation value is required. Earlier variants appeared in Carri`ere [7] and Tsitsiklis and Van Roy [22]. The regression-based ap- proach is the focus of this chapter and we collectively refer to the various forms of regression estimators as LSM estimators.

3_{With kind permission of Springer Science+Business Media, partial results published in [14] are reproduced}

in this chapter. Most results of this chapter appear in an independent paper that has been submitted for publication [13].

Monte Carlo simulation is generally regarded as a method to overcome the curse of dimen- sionality because its convergence rate is typically independent of the number of state variables. Moreover, it is flexible enough to handle different types of models and payoff structures. How- ever, its convergence rate can be slow. The convergence rate of a biased and consistent Monte Carlo estimator such as an LSM estimator can be enhanced in two directions, variance reduc- tion or bias reduction. Although variance reduction for pricing American options by simulation has been widely studied (see for example [2, 6, 12, 15, 17]), little work has been done on reduc- ing the bias in the estimators of American option values. Older bias-correction techniques in the literature are the ones that result in significantly increased estimator variance and/or com- putational time (see Broadie and Glasserman [4], and Carri`ere [7]). An exception is the one suggested by Avramidis and Hyden [2], which recursively averages the stochastic mesh high- and low-biased estimator at each exercise opportunity.

Recent works by Whitehead et al. [24, 25, 26] have made a remarkable contribution to bias reduction for Monte Carlo pricing of early-exercise options. They introduce a general bias-reduction technique, based on large sample theory, that corrects stochastic tree estima- tors. This technique corrects the bias due to making incorrect exercise decisions, which is the major source of bias in the estimators. In [25], numerical results from a well-studied multi- variate pricing problem demonstrate that this technique significantly reduces estimator bias, hence increasing the convergence rate. A rigorous proof justifying the bias-reduction method for the high-biased stochastic tree estimator is given in [24, 26]. Motivated by these results, bias-reduced stochastic mesh estimators are given in Kan et al. [14] and Whitehead [24]. Though not yet rigorously justified, exploratory numerical results in [14, 24] clearly show the effectiveness of the technique in removing the bias from the mesh estimators.

This motivates the extension of the bias-reduction technique to LSM estimators. At each exercise opportunity we derive an approximation to estimator bias using a well-known dis- tributional approximation for least-squares regression estimators. The bias approximation is subtracted from the LSM estimator at each exercise opportunity resulting in a corrected esti-

mator. This is done for each of the high- and low-biased LSM estimators. The bias-corrected LSM estimators are tested on a well-studied multivariate pricing problem used in Broadie and Glasserman [6], specifically an American call option on the maximum of five underlying as- sets. Numerical results show that the bias-correction technique significantly reduces the bias of these two types of LSM estimators.

In [14] the bias approximation and the corrected version of the high-biased LSM estimator is presented and partial numerical results show the effectiveness of the corrected high- and low-biased estimators. In this chapter, we derive the bias approximation for the low-biased LSM estimator (not given in MCQMC [14] due to space constraints) and provide extensive numerical results supporting this approach to bias correction. To be complete, we also include the derivation of the bias approximation for the high-biased estimator in this chapter. While our numerical results show that bias correction is not particularly useful for the high-biased estimator, it is very effective for the low-biased estimator. By that we mean the reduction in relative error of the high-biased estimator is not large enough to be useful in improving the computational efficiency since the additional computing time for the bias approximation is not negligible. Therefore, Item 4 on the following list is valid for the low-biased estimator, but not for the high-biased estimator; all the other points hold for both estimators.

1. The bias reduction method

∙ is independent of the dimension;

∙ holds for very general asset-price process and payoff structures;

∙ is simple to implement by modifying a few lines of code in existing algorithms;

∙ has little impact on estimator variance; and

∙ results in a consistent corrected estimator under fairly general conditions (Corollary 3.2).

2. The numerical results displayed in Section 3.5 show that the corrected estimators are always better (as measured by price, relative error and root mean square error) than the

uncorrected versions across all combinations of option moneyness, number of exercise opportunities and sample size.

3. The relative superiority of the corrected to uncorrected estimators decreases with sample size due to estimator consistency — there is less bias to correct for large sample sizes.

4. By allowing for a better tradeoff between decreasing sample size Min exchange for an increased number of repeated valuations, the technique permits increased computational efficiencies over trivial parallel implementations of existing algorithms. The scale of this benefit is constrained mainly by the number of processors available. Nowadays, it is not uncommon for financial institutions to have thousands of processors available for pricing purposes (see Staum [19]).

It is worth mentioning that the bias-reduction method is designed to correct the LSM esti- mators to the true approximation value given by a set of basis functions used in the regression. In other words, the corrected LSM estimators suffer from the same problem as the uncorrected LSM estimators — they do not converge to the true option value with a finite number of basis functions. This is a separate problem that is beyond the scope of this chapter. We note, how- ever, that for a given set of basis functions the corrected estimators perform better than their uncorrected counterparts.

The remainder of this chapter is organized as follows. Section 3.2 presents the valuation framework underlying the American option pricing problem. Section 3.3 discusses how to construct high- and low-biased estimators using the least-squares Monte Carlo method. Section 3.4 decomposes the error and derives the bias approximations for the LSM estimators. A proof of convergence is also given for the corrected low-biased estimator. Section 3.5 provides numerical results showing the effectiveness of this technique. Section 3.6 concludes the chapter and outlines future work.