Convergence en loi de Dirichlet de certaines intégrales stochastiques

Récemment, Bouleau a proposé une extension du principe d'invariance fonctionnelle de Donsker qui met en évidence la convergence en loi de Dirichlet d'une marche aléatoire erronée vers la structure d'Ornstein-Uhlenbeck sur l'espace de Wiener. Le but de cet article est d'étendre ce résultat à certaines familles d'intégrales stochastiques.Principe d'invariance, intégrales stochastiques, formes de Dirichlet, opérateur carré du champ, domaine vectoriel, erreurs.

Error structures and parameter estimation

This article proposes a link between statistics and the theory of Dirichlet forms used to compute errors. The error calculus based on Dirichlet forms is an extension of classical Gauss' approach to error propagation. The aim of this paper is to derive error structures from measurements. The links with Fisher's information lay the foundations of a strong connection with experiment. We show that this connection behaves well towards changes of variables and is related to the theory of asymptotic statistics

Convergence en loi de Dirichlet de certaines intégrales stochastiques

URL des Cahiers : https://halshs.archives-ouvertes.fr/CAHIERS-MSECahiers de la Maison des Sciences Economiques 2005.36 - ISSN : 1624-0340Recently, Nicolas Bouleau has proposed an extension of the Donsker's invariance principle in the framework of Dirichlet forms. He proves that an erroneous random walk of i.i.d random variables converges in Dirichlet law toward the Ornstein-Uhlenbeck error structure on the Wiener space [4]. The aim of this paper is to extend this result to some families of stochastic integrals.Récemment, Bouleau a proposé une extension du principe d'invariance fonctionnelle de Donsker qui met en évidence la convergence en loi de Dirichlet d'une marche aléatoire erronée vers la structure d'Ornstein-Uhlenbeck sur l'espace de Wiener. Le but de cet article est d'étendre ce résultat à certaines familles d'intégrales stochastiques

On an extension of the Hilbertian central limit theorem to Dirichlet forms

Permanent link to this document: http://projecteuclid.org/euclid.ojm/1216151109International audienceIn a recent paper, Nicolas Bouleau provides a new tool, based on the language of Dirichlet forms, to study the propagation of errors and reinforce the historical approach of Gauss. In the same way that the practical use of the normal distribution in statistics may be explained by the central limit theorem, the aim of this paper is to underline the importance of a family of error structures by asymptotic arguments

Likelihood-Related Estimation Methods and Non-Gaussian GARCH Processes

This article discusses the finite distance properties of three likelihood-based estimation strategies for GARCH processes with non-Gaussian conditional distributions : (1) the maximum likelihood approach ; (2) the Quasi maximum Likelihood approach ; (3) a multi-steps recursive estimation approach (REC). We first run a Monte Carlo test which shows that the recursive method may be the most relevant approach for estimation purposes. We then turn to a sample of SP500 returns. We confirm that the REC estimates are statistically dominating the parameters estimated by the two other competing methods. Regardless of the selected model, REC estimates deliver the more stable results.Maximum likelihood method, related-GARCH process, recursive estimation method, mixture of Gaussian distribution, Generalized Hyperbolic distributions, SP500.

Martingalized Historical approach for Option Pricing

In a discrete time option pricing framework, we compare the empirical performance of two pricing methodologies, namely the affine stochastic discount factor and the empirical martingale correction methodologies. Using a CAC 40 options dataset, the differences are found to be small : the higher order moment correction involved in the SDF approach may not be that essential to reduce option pricing errors.Generalized hyperbolic distribution, option pricing, incomplete market, CAC 40, Stochastic Discount Factor, martingale correction.

Option Pricing under GARCH models with Generalized Hyperbolic distribution (II) : Data and Results

In this paper, we provide a new dynamic asset pricing model for plain vanilla options and we discuss its ability to produce minimum mispricing errors on equity option books. The data set is the daily log returns of the French CAC40 index, on the period January 2, 1988, October 26, 2007. Under the historical measure, we adjust, on this data set, an EGARCH model with Generalized Hyperbolic innovations. We have shown (Chorro, Guégan and Ielpo, 2008) that when the pricing kernel is an exponential affine function of the state variables, the risk neutral distribution is unique and implies again a Generalized Hyperbolic dynamic, with changed parameters. Thus, using this theoretical result associated to Monte Carlo simulations, we compare our approach to natural competitors in order to test its efficiency. More generally, our empirical investigations analyze the ability of specific parametric innovations to reproduce market prices in the context of the exponential affine specification of the stochastic discount factor.Generalized Hyperbolic Distribution, Option pricing, Incomplete market, CAC40.

Option pricing for GARCH-type models with generalized hyperbolic innovations

In this paper, we provide a new dynamic asset pricing model for plain vanilla options and we discuss its ability to produce minimum mispricing errors on equity option books. Given the historical measure, the dynamics of assets are modeled by Garch-type models with generalized hyperbolic innovations and the pricing kernel is an exponential affine function of the state variables, we show that the risk neutral distribution is unique and implies again a generalized hyperbolic dynamics with changed parameters. We provide an empirical test for our pricing methodology on two data sets of options respectively written on the French CAC 40 and the American SP 500. Then, using our theoretical result associated with Monte Carlo simulations, we compare this approach to natural competitors in order to test its efficiency. More generally, our empirical investigations analyze the ability of specific parametric innovations to reproduce market prices in the context of an exponential affine specification of the stochastic discount factor.Generalized hyperbolic distribution, option pricing, incomplete markets, CAC 40, SP 500, GARCH-type models.

Option Pricing under GARCH models with Generalized Hyperbolic innovations (I) : Methodology

In this paper, we present an alternative to the Black Scholes model for a discrete time economy using GARCH-type models for the underlying asset returns with Generalized Hyperbolic (GH) innovations that are potentially skewed and leptokurtic. Assuming that the stochastic discount factor is an exponential affine function of the states variables, we show that this class of distributions is stable under the Risk neutral change of probability.GARCH, Generalized Hyperbolic Distribution, pricing, risk neutral distribution.

Martingalized Historical approach for Option Pricing

In a discrete time option pricing framework, we compare the empirical performance of two pricing methodologies, namely the affine stochastic discount factor (SDF) and the empirical martingale correction methodologies. Using a CAC 40 options dataset, the differences are found to be small: the higher order moment correction involved in the SDF approach may not be that essential to reduce option pricing errors. This paper puts into evidence the fact that an appropriate modelling under the historical measure associated with an adequate correction (that we call here a ”martingale correction”) permits to provide option prices which are close to market ones.Generalized Hyperbolic Distribution; Option pricing; Incomplete market; CAC40; Stochastic Discount Factor; Martingale Correction