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

Abstract

We analyze the convergence of the Longstaff-Schwartz algorithm relying on only a single set of independent Monte Carlo sample paths that is repeatedly reused for all exercise time-steps. We prove new estimates on the stochastic component of the error of this algorithm whenever the approximation architecture is any uniformly bounded set of L 2 functions of finite Vapnik-Chervonenkis dimension (VC-dimension), but in particular need not necessarily be either convex or closed. We also establish new overall error estimates, incorporating bounds on the approximation error as well, for certain nonlinear, nonconvex sets of neural networks.

Discrete-type Approximations for Non-Markovian Optimal Stopping Problems: Part II

“…Assumptions (H1-H2) are well-known in the classical case of the Longstaff-Schwartz algorithm based on Markov chains. See e.g Egloff [12], Zanger [40,41] and other references therein. for 1 ≤ j ≤ e(k, T ) − 1, N ≥ 2 and k ≥ 1.…”

“…for 1 ≤ j ≤ e(k, T ) − 1, N ≥ 2 and k ≥ 1. There are some conditions for the existence of both minimizers as discussed in Remark 5.4 by Zanger [40], for instance, compactness of H k N,j . We observe that we may assume that S j k ; 1 ≤ j ≤ e(k, T ) they are compact because all the hitting times (T k n ) e(k,T ) n=1…”

“…The key assumptions in Egloff [12] require the architecture approximating sets designed to approach the continuation values to be closed, convex and uniformly bounded with a finite Vapnik-Chervonenkis dimension (VC-dimension). Later, Zanger [40,41,42] presents more general results by employing nonlinear approximating architecture spaces and not necessarily convex and closed. Sequential design schemes are introduced by Gramacy and Ludkovski [17] in order to infer the underlying optimal stopping boundary.…”

“…In order to prove convergence of the Monte Carlo scheme with explicit error estimates, we make use of statistical learning theory techniques originally employed very successfully by Egloff [12,13] and Zanger [40,41,42] based on Markov chain approximations for Markov diffusions. We show that the same philosophy employed by Zanger [40,41,42] can be used to prove convergence of a Longstaff-Schwartz-type algorithm associated with rather general classes of reward processes Z = F (X) where X is an R n -valued F-adapted continuous process. By means of statistical learning theory techniques, we get precise error estimates (w.r.t N ) for each discretization level k encoded by ǫ k .…”

Discrete-type approximations for non-Markovian optimal stopping problems: Part I

“…The main obstacle is the obtention of continuation values (expressed as F ti -conditional expectations) in the dynamic programming algorithm. In this case, least-squares Monte-Carlo methods can be employed by using non-parametric regression techniques based on a suitable choice of regression polynomials (see e.g [34] and other references therein). Another popular approach is to make use of some available representations for conditional expectations in terms of a suitable ratio of unconditional expectations obtained by using Malliavin calculus (see e.g [12]).…”