Efficiency improvements for pricing American options with a stochastic mesh

Abstract

We develop and study general-purpose techniques for improving the e ciency of the stochastic mesh method that was recently developed for pricing American options via Monte Carlo simulation. First, we d e v elop a mesh-based, biased-low estimator. By recursively averaging the low and high estimators at each stage, we obtain a signi cantly more accurate point estimator at each of the mesh points. Second, we adapt the importance sampling ideas for simulation of European path-dependent options in Glasserman, Heidelberger, and Shahabuddin (1998a) to pricing of American options with a stochastic mesh. Third, we s k etch generalizations of the mesh method and we discuss links with other techniques for valuing American options. Our empirical results show that the bias-reduced point estimates are much more accurate than the standard mesh-method point estimates. Importance sampling is found to increase accuracy for a smooth optionpayo functions, while variance increases are possible for non-smooth payo s.

Simulation-based pricing of convertible bonds

“…and propose a method for calculating prices of American-style options with simulated trees that generate two estimates, one biased high and one biased low. Broadie and Glasserman (2004), Broadie et al, (2000), and Avramidis and Hyden (1999) develop stochastic-mesh methods with different choices for mesh weights. Finally, Broadie and Cao (2003), Haugh and Kogan (2004), and Rogers (2002) suggest a simulation method that uses a duality approach for pricing Bermudan options.…”

Valuation of the Reset Options Embedded in Some Equity-Linked Insurance Products

“…This is in contrast to the biased low Broadie-Glasserman path estimator, which requires generating an independent set of trajectories, thus increasing the computational effort. This feature is exploited by Avramidis and Hyden (1999a) for proposing a so-called average mesh estimator, Q a . Instead of recursively computing biased high and biased low estimators separately, they suggest computing a revised estimator by averaging the biased high and biased low estimators at each backward induction step.…”

“…In fact, the confidence interval in Equation (13) will be conservative. In our numerical comparison, the above confidence interval is not constructed because we are primarily interested in assessing the bias arising from our proposed high-biased estimators and comparing that to the average mesh estimator of Avramidis and Hyden (1999a).…”

“…The first is the high biased Broadie-Glasserman (BG) mesh estimator, Q h (0, S 0 ). The second is the average mesh estimator, Q a (0, S 0 ), considered by Avramidis and Hyden (1999a). We label this estimator as the AH mesh estimator.…”

“…For this method to be of practical use, one must incorporate extensive variance reduction techniques, as suggested by Broadie and Glasserman (1997b). • With 10 exercise points, the average or AH mesh estimators proposed by Avramidis and Hyden (1999a) appear to be effective at reducing the bias of the BG mesh estimators. However, as we consider more frequent exercise points, this method starts to show signs of deterioration.…”