This paper studies a classical extension of the Black and Scholes model for option pricing, often known as the Hull and White model. Our specification is that the volatility process is assumed not only to be stochastic, but also to have long-memory features and properties. We study here the implications of this continuous-time long-memory model, both for the volatility process itself as well as for the global asset price process. We also compare our model with some discrete time approximations. Then the issue of option pricing is addressed by looking at theoretical formulas and properties of the implicit volatilities as well as statistical inference tractability. Lastly, we provide a few simulation experiments to illustrate our results. Copyright Blackwell Publishers Inc 1998.

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The authors consider the problem of estimating the density g of independent and identically distributed variables X i , from a sample 21,. . . , 2, such that 2; = Xi + QE; for i = 1,. . . , n, and E is noise independent of X, with QE having a known distribution. They present a model selection procedure allowing one to construct an adaptive estimator of g and to find nonasymptotic risk bounds. The estimator achieves the minimax rate of convergence, in most cases where lower bounds are available. A simulation study gives an illustration of the good practical performance of the method. Deconvolution adaptative de densite par contraste penaliseRbumt! : Les auteurs considkrent le probltme de dkonvolution, c'est-&-dire de I'estimation de la densitt de variables altatoires identiquement distributes X,, ii partir de I'observation de Zi, oh 2; = X , + QE, pour i = 1,. . . , n, et oh les erreurs QE; sont de densit6 connue. Par une proc6dure de selection de modtles qui permet d'obtenir des bornes de risque non asymptotiques, ils construisent un estimateur adaptatif de la densitt des X ; . L'estimateur atteint automatiquement la vitesse minimax dans la plupart des cas, que les erreurs ou la densit6 & estimer soient peu ou trhs dgulitres. Une ttude par simulation illustre les bonnes performances pratiques de la mtthode.

We study the model Y D X C ". We assume that we have at our disposal independent identically distributed observations Y 1 ,...,Y n and " 1 ,...," M . The .X j / 1 j n are independent identically distributed with density f , independent of the ." j / 1 j n , independent identically distributed with density f " . The aim of the paper is to estimate f without knowing f " . We first define an estimator, for which we provide bounds for the integrated L 2 -risk.We consider ordinary smooth and supersmooth noise " with regard to ordinary smooth and supersmooth densities f. Then we present an adaptive estimator of the density of f. This estimator is obtained by penalization of a projection contrast and yields to model selection. Lastly, we present simulation experiments to illustrate the good performances of our estimator and study from the empirical point of view the importance of theoretical constraints.

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By fractional integration of a square root volatility process, we propose in this paper a long memory extension of the Heston (Rev Financ Stud 6:327-343, 1993) option pricing model. Long memory in the volatility process allows us to explain some option pricing puzzles as steep volatility smiles in long term options and co-movements between implied and realized volatility. Moreover, we take advantage of the analytical tractability of affine diffusion models to clearly disentangle long term components and short term variations in the term structure of volatility smiles. In addition, we provide a recursive algorithm of discretization of fractional integrals in order to be able to implement a method of moments based estimation procedure from the high frequency observation of realized volatilities.

We consider a one-dimensional diffusion process (Xt) which is observed at n + 1 discrete times with regular sampling interval ∆. Assuming that (Xt) is strictly stationary, we propose nonparametric estimators of the drift and diffusion coefficients obtained by a penalized least squares approach. Our estimators belong to a finite-dimensional function space whose dimension is selected by a data-driven method. We provide non-asymptotic risk bounds for the estimators. When the sampling interval tends to zero while the number of observations and the length of the observation time interval tend to infinity, we show that our estimators reach the minimax optimal rates of convergence. Numerical results based on exact simulations of diffusion processes are given for several examples of models and illustrate the qualities of our estimation algorithms. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2007, Vol. 13, No. 2, 514-543. This reprint differs from the original in pagination and typographic detail.

International audienceIn this paper, we consider a multidimensional convolution model for which we provide adaptive anisotropic kernel estimators of a signal density $f$ measured with additive error. For this, we generalize Fan's~(1991) estimators to multidimensional setting and use a bandwidth selection device in the spirit of Goldenschluger and Lepski's~(2011) proposal fr density estimation without noise. We consider first the pointwise setting and then, we study the integrated risk. Our estimators depend on an automatically selected random bandwidth. We assume both ordinary and super smooth components for measurement errors, which have known density. We also consider both anisotropic H\"{o}lder and Sobolev classes for $f$. We provide non asymptotic risk bounds and asymptotic rates for the resulting data driven estimator, which is proved to be adaptive. We provide an illustrative simulation study, involving the use of Fast Fourier Transform algorithms. We conclude by a proposal of extension of the method to the case of unknown noise density, when a preliminary pure noise sample is available