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

A Radon-Nikodym derivative for almost subadditive set functions

In classical measure theory, the Radon-Nikodym theorem states in a concise condition, namely domination, how a measure can be factorized by another (bounded) measure through a density function. Several approaches have been undertaken to see under which conditions an exact factorization can be obtained with set functions that are not σ-additive (for instance finitely additive set functions or submeasures). We provide a Radon-Nikodym type theorem with respect to a measure for almost subadditive set functions of bounded sum. The necessary and sufficient condition to guarantee a one-sided Radon-Nikodym derivative remains the standard domination condition for measures.

Option Valuation with Conditional Heteroskedasticity and Non-Normality

We provide results for the valuation of European style contingent claims for a large class of specifications of the underlying asset returns. Our valuation results obtain in a discrete time, infinite state-space setup using the no-arbitrage principle and an equivalent martingale measure. Our approach allows for general forms of heteroskedasticity in returns, and valuation results for homoskedastic processes can be obtained as a special case. It also allows for conditional non-normal return innovations, which is critically important because heteroskedasticity alone does not suffice to capture the option smirk. We analyze a class of equivalent martingale measures for which the resulting risk-neutral return dynamics are from the same family of distributions as the physical return dynamics. In this case, our framework nests the valuation results obtained by Duan (1995) and Heston and Nandi (2000) by allowing for a time-varying price of risk and non-normal innovations. We provide extensions of these results to more general equivalent martingale measures and to discrete time stochastic volatility models, and we analyze the relation between our results and those obtained for continuous time models. Nous présentons les résultats d’une étude portant sur l’évaluation de créances éventuelles de style européen pour une grande variété de caractéristiques liées au rendement des actifs sous-jacents. Les résultats de notre évaluation proposent en temps discret une formule état-espace infinie, à partir du principe de non-arbitrage et d’une mesure de martingale équivalente. Notre approche permet de tenir compte de formes générales d’hétéroscédasticité dans les rendements et d’obtenir, dans des cas spéciaux, des résultats d’évaluation liés aux processus homoscédastiques. Elle permet aussi de considérer les innovations conditionnellement non normales en matière de rendement, ce qui représente un facteur critique, compte tenu du fait que l’hétéroscédasticité ne permet pas, à elle seule, de saisir pleinement le caractère ironique de l’option. Nous analysons une catégorie de mesures de martingale équivalentes dont la dynamique du rendement risque-neutre obtenu est de la même famille de distribution que la dynamique du rendement physique. Dans ce cas, notre cadre d’étude soutient les résultats d’évaluation obtenus par Duan (1995) et par Heston et Nandi (2000) et tient compte du coût du risque variant dans le temps et des innovations non normales. Nous étendons ces résultats aux mesures de martingale équivalentes plus générales et aux modèles de volatilité stochastique en temps discret et analysons aussi la relation entre nos résultats et ceux obtenus dans le cas des modèles en temps continu.GARCH, risk-neutral valuation, no-arbitrage, non-normal innovations, GARCH (hétéroscédasticité conditionnelle autorégressive généralisée), évaluation du risque neutre, absence d’arbitrage, innovations non normales

Hilbert C*-modules and amenable actions

We study actions of discrete groups on Hilbert $C^*$-modules induced from topological actions on compact Hausdorff spaces. We show non-amenability of actions of non-amenable and non-a-T-menable groups, provided there exists a quasi-invariant probability measure which is sufficiently close to being invariant.Comment: Final version, to appear in Studia Mathematic

On the use of reproducing kernel Hilbert spaces in functional classification

The H\'ajek-Feldman dichotomy establishes that two Gaussian measures are either mutually absolutely continuous with respect to each other (and hence there is a Radon-Nikodym density for each measure with respect to the other one) or mutually singular. Unlike the case of finite dimensional Gaussian measures, there are non-trivial examples of both situations when dealing with Gaussian stochastic processes. This paper provides: (a) Explicit expressions for the optimal (Bayes) rule and the minimal classification error probability in several relevant problems of supervised binary classification of mutually absolutely continuous Gaussian processes. The approach relies on some classical results in the theory of Reproducing Kernel Hilbert Spaces (RKHS). (b) An interpretation, in terms of mutual singularity, for the "near perfect classification" phenomenon described by Delaigle and Hall (2012). We show that the asymptotically optimal rule proposed by these authors can be identified with the sequence of optimal rules for an approximating sequence of classification problems in the absolutely continuous case. (c) A new model-based method for variable selection in binary classification problems, which arises in a very natural way from the explicit knowledge of the RN-derivatives and the underlying RKHS structure. Different classifiers might be used from the selected variables. In particular, the classical, linear finite-dimensional Fisher rule turns out to be consistent under some standard conditions on the underlying functional model