12 research outputs found
B-spline techniques for volatility modeling
This paper is devoted to the application of B-splines to volatility modeling,
specifically the calibration of the leverage function in stochastic local
volatility models and the parameterization of an arbitrage-free implied
volatility surface calibrated to sparse option data. We use an extension of
classical B-splines obtained by including basis functions with infinite
support. We first come back to the application of shape-constrained B-splines
to the estimation of conditional expectations, not merely from a scatter plot
but also from the given marginal distributions. An application is the Monte
Carlo calibration of stochastic local volatility models by Markov projection.
Then we present a new technique for the calibration of an implied volatility
surface to sparse option data. We use a B-spline parameterization of the
Radon-Nikodym derivative of the underlying's risk-neutral probability density
with respect to a roughly calibrated base model. We show that this method
provides smooth arbitrage-free implied volatility surfaces. Finally, we sketch
a Galerkin method with B-spline finite elements to the solution of the partial
differential equation satisfied by the Radon-Nikodym derivative.Comment: 25 page
Partial functional quantization and generalized bridges
In this article, we develop a new approach to functional quantization, which consists in discretizing only a finite subset of the Karhunen-Loève coordinates of a continuous Gaussian semimartingale . Using filtration enlargement techniques, we prove that the conditional distribution of knowing its first Karhunen-Loève coordinates is a Gaussian semimartingale with respect to a bigger filtration. This allows us to define the partial quantization of a solution of a stochastic differential equation with respect to by simply plugging the partial functional quantization of in the SDE. Then we provide an upper bound of the -partial quantization error for the solution of SDEs involving the -partial quantization error for , for . The convergence is also investigated. Incidentally, we show that the conditional distribution of a Gaussian semimartingale , knowing that it stands in some given Voronoi cell of its functional quantization, is a (non-Gaussian) semimartingale. As a consequence, the functional stratification method developed in [6] amounted, in the case of solutions of SDEs, to using the Euler scheme of these SDEs in each Voronoi cell
A fast nearest neighbor search algorithm based on vector quantization
In this article, we propose a new fast nearest neighbor search algorithm,
based on vector quantization. Like many other branch and bound search
algorithms [1,10], a preprocessing recursively partitions the data set into
disjointed subsets until the number of points in each part is small enough. In
doing so, a search-tree data structure is built. This preliminary recursive
data-set partition is based on the vector quantization of the empirical
distribution of the initial data-set. Unlike previously cited methods, this
kind of partitions does not a priori allow to eliminate several brother nodes
in the search tree with a single test. To overcome this difficulty, we propose
an algorithm to reduce the number of tested brother nodes to a minimal list
that we call "friend Voronoi cells". The complete description of the method
requires a deeper insight into the properties of Delaunay triangulations and
Voronoi diagram
Functional quantization-based stratified sampling methods
In this article, we propose several quantization-based stratified sampling methods to reduce the variance of a Monte Carlo simulation. Theoretical aspects of stratification lead to a strong link between optimal quadratic quantization and the variance reduction that can be achieved with stratified sampling. We first put the emphasis on the consistency of quantization for partitioning the state space in stratified sampling methods in both finite and infinite dimensional cases. We show that the proposed quantization-based strata design has uniform efficiency among the class of Lipschitz continuous functionals. Then a stratified sampling algorithm based on product functional quantization is proposed for path-dependent functionals of multi-factor diffusions. 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 facto
Jupyter Notebooks – a publishing format for reproducible computational workflows
It is increasingly necessary for researchers in all fields to write computer code, and in order to reproduce research results, it is important that this code is published. We present Jupyter notebooks, a document format for publishing code, results and explanations in a form that is both readable and executable. We discuss various tools and use cases for notebook documents
Quelques aspects de la quantification optimale et applications à la finance
This thesis is concerned with the study of optimal quantization and its applications. We deal with theoretical, algorithmic and numerical aspects. It is composed of five chapters. In the first section, we study the link between the stratification-based variance reduction and optimal quantization. In the case where the considered random variable is a Gaussian process, a simulation scheme with a linear cost is proposed for the conditional distribution of the process in a stratum. The second chapter is devoted to the numerical approximation of the Karhunen-Loève eigensystem of a Gaussian process with the so-called Nyström method. In the third section, we propose a new approach for the quantization of the solutions of SDE, whose convergence is investigated. These results lead to a new cubature scheme for the solutions of stochastic differential equations, which is developed in the fourth chapter, and which is tested with option pricing problems. In the last chapter, we present a new tree-based fast nearest neighbor search algorithm, based on the quantization of the empirical distribution of the considered data set.Cette thèse est consacrée à l'étude de la quantification optimale et ses applications. Nous y abordons des aspects théoriques, algorithmiques et numériques. Elle comporte cinq chapitres. Dans la première partie, nous étudions liens entre la réduction de variance par stratification et la quantification optimale. Dans le cas ou la variable aléatoire considérée est un processus Gaussien, un schéma de simulation de complexité linéaire est développé pour la loi conditionnelle à une strate du processus en question. Le second chapitre est consacré à l'évaluation numérique de la base de Karhunen-Loève d'un processus Gaussien par la méthode de Nyström. Dans la troisième partie, nous proposons une nouvelle approche de la quantification de solutions d'EDS, dont nous étudions la convergence. Ces résultats conduisent à un nouveau schéma de cubature pour les solutions d'équations différentielles stochastiques, qui est développé dans le quatrième chapitre, et que nous éprouvons sur des problèmes de valorisation d'options. Dans le cinquième chapitre, nous présentons un nouvel algorithme de recherche rapide de plus proche voisin par arbre, basé sur la quantification de la loi empirique du nuage de points considéré
Some aspects of optimal quantization and applications to finance
PARIS-BIUSJ-Mathématiques rech (751052111) / SudocSudocFranceF