Functional quantization-based stratified sampling methods

Abstract

In this article, we propose several quantization-based strati ed sampling methods to reduce the variance of a Monte Carlo simulation. Theoretical aspects of strati cation lead to a strong link between optimal quadratic quantization and the variance reduction that can be achieved with strati ed sampling. We rst put the emphasis on the consistency of quantization for partitioning the state space in strati ed sampling methods in both nite and in nite-dimensional cases. We show that the proposed quantization-based strata design has uniform e ciency among the class of Lipschitz continuous functionals. Then a strati ed sampling algorithm based on product functional quantization is proposed for path-dependent functionals of multifactor di usions. The method is also available for other Gaussian processes such as Brownian bridge or Ornstein-Uhlenbeck processes. We derive in detail the case of Ornstein-Uhlenbeck processes. We also study the balance between the algorithmic complexity of the simulation and the variance reduction factor.

Small-Time Asymptotics for Gaussian Self-Similar Stochastic Volatility Models

“…There exist explicit formulas for the Karhunen-Loève characteristics of various Gaussian processes. For Brownian motion, Brownian bridge, and OU processes, such formulas can be found in [16]. For OU bridges, one can consult [17,15], and for the Gaussian process introduced in [18], the Karhunen-Loève decomposition can be found in the same paper.…”

“…Let S be the asset price process in the model considered in (16). Define the call and the put pricing functions by…”

“…Next, we turn our attention to the out-of-the-money put pricing function T → P (T ) with 0 < K < s 0 . The asymptotic behavior of the put pricing function with 0 < K < s 0 will be characterized using the symmetry properties of the model in (16). In ( [41], Lemma 9.25), several equivalent conditions are given for the symmetry of a stochastic volatility model.…”

Introduction to vector quantization and its applications for numerics

“…[59] for applications to the pricing and hedging of path-dependent options. Let us cite as well applications to variance reduction by universal stratified sampling (see [18]). But we will not go further in that direction in this paper.…”

Multifractional Stochastic Volatility Models

“…Functional quantization of Gaussian processes has become an active field of research in recent years since the seminal article Luschgy and Pagès (2002). As far as applications are concerned, cubature methods (Pagès and Printems 2005a;Corlay 2010b) and variance reduction methods (Corlay and Pagès 2010;Lejay and Reutenauer 2008) based on functional quantization have been proposed. However, as the numerical use of functional quantizers requires the evaluation of the Karhunen-Loève eigenfunctions, this method was restricted to processes for which a closed-form expression for this expansion is known, such as Brownian motion.…”