Introduction to vector quantization and its applications for numerics

Pagès, Gilles

doi:10.1051/proc/201448002

Cited by 65 publications

(58 citation statements)

References 57 publications

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“…In this section we introduce optimal quadratic quantization of a random variable (also known as vector quantization), which will be a necessary tool toward a specific discretization of a stochastic process, known as recursive marginal quantization (henceforth RMQ). We refer to Graf and Luschgy (2000) and Pagès (2015) for vector quantization and to Pagès and Sagna (2015) for the first paper on RMQ 1 .…”

Section: Essentials On Quantizationmentioning

confidence: 99%

“…When applied to random vectors, quantization provides the best, according to a distance that is commonly measured using the Euclidean norm, possible approximation to the original distribution via a discrete random vector taking a finite number of values. Many numerical procedures have been studied to obtain optimal quadratic quantizers of random vectors even in high dimension and most of them are based on stochastic optimization algorithms, see Pagès (2015), which are typically very time consuming. Very recently vector quantization has been applied to recursively discretize stochastic processes.…”

Section: Introductionmentioning

confidence: 99%

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Quantization Goes Polynomial

2017

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Quantization algorithms have been recently successfully adopted in option pricing problems to speed up Monte Carlo simulations thanks to the high convergence rate of the numerical approximation. In particular, recursive marginal quantization has been proven a flexible and versatile tool when applied to stochastic volatility processes. In this paper we apply for the first time these techniques to the family of polynomial processes, by exploiting, whenever possible, their peculiar properties. We derive theoretical results to assess the approximation errors, and we describe in numerical examples practical tools for fast exotic option pricing. JEL classification codes: C63, G13.

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Section: Essentials On Quantizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantization Goes Polynomial

2017

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“…However, computationally, finding explicitly Γ * can be a challenging task. This has motivated the introduction of sub-optimal criteria linked to the notion of stationary quantizer [37]:…”

Section: Optimal Quantizationmentioning

confidence: 99%

A Backward Monte Carlo Approach to Exotic Option Pricing

et al. 2015

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We propose a novel algorithm which allows to sample paths from an underlying price process in a local volatility model and to achieve a substantial variance reduction when pricing exotic options. The new algorithm relies on the construction of a discrete multinomial tree. The crucial feature of our approach is that -in a similar spirit to the Brownian Bridge -each random path runs backward from a terminal fixed point to the initial spot price. We characterize the tree in two alternative ways: in terms of the optimal grids originating from the Recursive Marginal Quantization algorithm and following an approach inspired by the finite difference approximation of the diffusion's infinitesimal generator. We assess the reliability of the new methodology comparing the performance of both approaches and benchmarking them with competitor Monte Carlo methods.JEL codes: C63, G12, G13

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“…where ϕ is the density of X, λ d is the Lebesgue measure on R d and r J p,d " inf N ě1 N 1 d }Uṕ U N } p , U L " U`p0, 1q d˘. For more insights on the mathematical/probabilistic aspects of Optimal quantization theory, we refer to [GL00,Pag15].…”

Section: Introductionmentioning

confidence: 99%

New weak error bounds and expansions for optimal quantization

Lemaire

Montes²,

Pagès³

2020

Journal of Computational and Applied Mathematics

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We propose new weak error bounds and expansion in dimension one for optimal quantization-based cubature formula for different classes of functions, such that piecewise affine functions, Lipschitz convex functions or differentiable function with piecewise-defined locally Lipschitz or α-Hölder derivatives. This new results rest on the local behaviours of optimal quantizers, the L r -L s distribution mismatch problem and Zador's Theorem. This new expansion supports the definition of a Richardson-Romberg extrapolation yielding a better rate of convergence for the cubature formula. An extension of this expansion is then proposed in higher dimension for the first time. We then propose a novel variance reduction method for Monte Carlo estimators, based on one dimensional optimal quantizers.

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Introduction to vector quantization and its applications for numerics

Cited by 65 publications

References 57 publications

Quantization Goes Polynomial

Quantization Goes Polynomial

A Backward Monte Carlo Approach to Exotic Option Pricing

New weak error bounds and expansions for optimal quantization

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