Pricing Bermudan Options via Multilevel Approximation Methods

Abstract

In this article we propose a novel approach to reducing the computational complexity of various approximation methods for pricing discrete time American or Bermudan options. Given a sequence of continuation values estimates corresponding to different levels of spatial approximation, we propose a multilevel low biased estimate for the price of the option. It turns out that the resulting complexity gain can be of order ε −1 with ε denoting the desired precision. The performance of the proposed multilevel algorithms is illustrated by a numerical example.

Faster comparison of stopping times by nested conditional Monte Carlo

“…Our method is used to enhance the efficiency of this control variate. For two variance reduction methods of this type, a Quasi-Control Variate as introduced in [9] and the Multilevel algorithm of [4], we demonstrate that Nested Conditional Monte Carlo can lead to sizeable efficiency gains. In fact, for both algorithms the additional variance reduction through incorporating our method is at least as large as the variance reduction of the original algorithm.…”

Convergence of a Least‐squares Monte Carlo Algorithm for American Option Pricing With Dependent Sample Data

“…Global regression methods involve minimizing a standard sum of squared differences empirical errors. In contrast, Belomestny () and Belomestny, Dickmann, and Nagapetyan () apply local polynomial kernel regression to the problem of pricing Bermudan options by means of Monte Carlo regression. In local polynomial kernel regression (see also Gyorfi et al.…”

“…Applying local polynomial kernel regression and also simulating a new, independent set of Monte Carlo sample trajectories (in addition to the samples used in the prior regression step), Belomestny () averages the corresponding payoffs stopped according to a simple (generally suboptimal) stopping rule for these paths to generate a low‐biased estimate for the option price that can be shown to possess improved convergence properties if a number of technical conditions hold (see Remark below). Belomestny, Dickmann, and Nagapetyan () extend the analysis of Belomestny () by examining the computational complexity of the algorithm in Belomestny (). They find the asymptotic complexity to be of order larger than ε 3 , where corresponds to the algorithm precision, but that it can be improved to order ε 2 in some cases using a related multilevel Monte Carlo regression approach.…”

“…Global regression methods involve minimizing a standard sum of squared differences empirical errors. In contrast, Belomestny (2011) and Belomestny, Dickmann, and Nagapetyan (2015) apply local polynomial kernel regression to the problem of pricing Bermudan options by means of Monte Carlo regression. In local polynomial kernel regression (see also Gyorfi et al 2002;Kohler 2010), an additional multiplicative factor involving a suitable kernel function is introduced into the sum-of-squared-differences regression minimization problem, and the approximation architecture is defined by means of an appropriate space of polynomials.…”

Multilevel Simulation Based Policy Iteration for Optimal Stopping--Convergence and Complexity

“…A detailed probabilistic treatment of a class of policy iterations (that includes Howard's one as a special case) as well as the description of the corresponding Monte Carlo algorithms is provided in Kolodko and Schoenmakers (2006). In the spirit of Belomestny and Schoenmakers (2011) (see also Belomestny et al (2013) and Bujok et al (2012)) we here develop a multilevel estimator, where the multilevel concept is applied to the number of inner Monte Carlo simulations needed to construct a new policy, rather than the discretization step size of a particular SDE as in Giles (2008). In this context we give a detailed analysis of the bias rates and the related variance rates that are crucial for the performance of the multilevel algorithm.…”