Columbia University
Monte Carlo Methods for Pricing American Options: Overview and New Results
An American option
allows the holder to choose the time
of exercise, so valuing such an option entails solving
an optimal stopping problem. Deterministic numerical methods
are generally inapplicable to high-dimensional pricing problems
(such as options depending on many underlying assets), making
Monte Carlo methods potentially attractive. But the embedded
optimal stopping problem complicates the valuation of American options
through simulation. The first part of this talk will be an overview of
methods developed in recent years to address this problem. These methods
apply weighted backward induction to simulated paths, with
weights defined through likelihood ratios, through calibration,
or implicitly through regression. The second part of this talk analyzes
conditions for convergence as both the number of paths
and number of basis functions for regression grow.
Using polynomials in the regressions, the number of
paths must grow exponentially with the number of basis
functions to ensure convergence when applied to Brownian
motion, faster when applied to geometric Brownian motion.
This analysis is based on joint work with Bin Yu.
of exercise, so valuing such an option entails solving
an optimal stopping problem. Deterministic numerical methods
are generally inapplicable to high-dimensional pricing problems
(such as options depending on many underlying assets), making
Monte Carlo methods potentially attractive. But the embedded
optimal stopping problem complicates the valuation of American options
through simulation. The first part of this talk will be an overview of
methods developed in recent years to address this problem. These methods
apply weighted backward induction to simulated paths, with
weights defined through likelihood ratios, through calibration,
or implicitly through regression. The second part of this talk analyzes
conditions for convergence as both the number of paths
and number of basis functions for regression grow.
Using polynomials in the regressions, the number of
paths must grow exponentially with the number of basis
functions to ensure convergence when applied to Brownian
motion, faster when applied to geometric Brownian motion.
This analysis is based on joint work with Bin Yu.