Quelques inégalités sur les fonctions monotones

Abstract

Dans les lignes qui suivent nous allons donner quelques suppléments à la “note sur quelques inégalites et formules d'approximation” de ce journal. 1 En partant de l'inégalité bien connue de Hölder-Jensen sur les fonctions convexes nous allons — à l'aide de la définition de convexité que j'avais introduit dans le mémoir cité — trouver une inégalité ayant une relation étroite avec la formule de moyenne de Weierstrass 2, où f(t) et ? (t) sont deux fonctions continues, la fonction ? (t) étant décroissante lorsque x crîlt de a à b.     

Note sur quelques inégalités et formules d'approximation

Abstract

Les calcuis pénibles et laborieux que nécessitent trop souvent les opérations d'assurance sur la vie, rendent fort désireux de trouver des formules d'approximation ou bien des inégalités générales, faisant connaître en chaque cas particulier, et avec le moins de travail possible, une limite supérieure ou inférieure des expressions dont on cherche la valeur numérique. Cela est évident surtout au cas où l'on désire senlement faire une évaluation “à peu près”. Les exemples en abondent.     

Le calcul de la prime en réassurance d'excédent de sinistres

Abstract

Deux des rapports traitant la réassurance en excédent de sinistres présentés au XI^ème Congrès International d'Actuaires à Paris, par M. I var Hesselberg et par M. P. Dubois, ont abouti à des conclusions si divergentes au sujet de l'applicabilité de la méthode qu'il peut y avoir intérêt à les examiner de plus près.     

Tangent Lévy market models

In this paper, we introduce a new class of models for the time evolution of the prices of call options of all strikes and maturities. We capture the information contained in the option prices in the density of some time-inhomogeneous Lévy measure (an alternative to the implied volatility surface), and we set this static code-book in motion by means of stochastic dynamics of It?’s type in a function space, creating what we call a tangent Lévy model. We then provide the consistency conditions, namely, we show that the call prices produced by a given dynamic code-book (dynamic Lévy density) coincide with the conditional expectations of the respective payoffs if and only if certain restrictions on the dynamics of the code-book are satisfied (including a drift condition à la HJM). We then provide an existence result, which allows us to construct a large class of tangent Lévy models, and describe a specific example for the sake of illustration.     

Time consistency of Lévy models

Time consistency of the models used is an important ingredient to improve risk management. The empirical investigation in this article gives evidence for some models driven by Lévy processes to be highly consistent. This means that they provide a good statistical fit of empirical distributions of returns not only on the timescale used for calibration but on various other timescales as well. As a result these models produce more reliable risk numbers and derivative prices.     

Skewness premium with Lévy processes

We study the skewness premium (SK) introduced by Bates [J. Finance, 1991, 46(3), 1009–1044] in a general context using Lévy processes. Under a symmetry condition, Fajardo and Mordecki [Quant. Finance, 2006, 6(3), 219–227] obtained that SK is given by Bates' x% rule. In this paper, we study SK in the absence of that symmetry condition. More exactly, we derive sufficient conditions for the excess of SK to be positive or negative, in terms of the characteristic triplet of the Lévy process under a risk-neutral measure.     

Numerical methods for Lévy processes

We survey the use and limitations of some numerical methods for pricing derivative contracts in multidimensional geometric Lévy models.      

Sur une loi de probabilité considérée par J. F. Steffensen

Abstract

Dans ce même périodique, vous avez considéré¹, à la page 7, la loi de probabilité de deux variables aléatoires X, Y,² où la probabilité élémentaire ?(x, y) dx dy pour que X et Y soient respectivement compris entre x et x + dx, y et y + dy, est de la forme où K, a ₁, a ₂, b ₁, b ₂ sont des constantes. Nous nous proposons, dans ce qui suit, d'apporter quelques compléments à votre exposé.     

Why is management a cliché?

This article introduces the term demotics of management by asking why so much management literature reads like a cliché. Typically this question has been approached by seeing the cliché as strategic. This article instead views the cliché as symptomatic. It marks a growing problem—how can management track labor out of the workplace and into the realm of social reproduction, a realm that is increasingly, with the tendency of immaterial labor, directly productive. This problem has produced not only the explosion of popular management literature, particularly in the United States, in the last 20 years, but also what might be called a demotics of management. This term may be understood as the proliferation of places where labor-power might be found. But the term also names management’s growing limits when faced with the dispersion and intensification of what Marx called the social individual. Management cannot adequately measure the labor it finds, and therefore must resort to miraculating what it encounters, to use a concept from the collaboration of Gilles Deleuze and Felix Guattari. The crisis of measurement brought on by the growing dominance of what Marx named the General Intellect is a profound challenge not only to management but specifically to its recording-machine, accounting. With the mounting crisis in recording, the very frenzy of popular management discourse reveals more than ever the threat that social reproductive labor already possesses the sociality to value itself differently and independently. The demotics of management thus marks both the proliferation of management in daily life, but also the prior, or what C.L.R. James might have called the completed, organization of immaterial labor that management encounters. Left to repeat what is already completed management can only utter the cliché, however, manically. At the same time, the cliché in management literature does mark the accomplishment of the circuit of exchange value, as it has since Taylor, even as it hints at the possibility of the separation from capital of a mass intellectuality where socialized labor may come to account for itself.     

Sur certains mouvements aléatoires discontinus

1. Introduction. Divers problèmes de Physique et de Calcul des Probabilités conduisent à des sehémas de mouvements aléatoires, régis par l'équation de Chapman, dans lesquels un certain point mobile ne se déplaee qu'un nombre fini de fois et reste immobile entre les instants aléatoires auxquels ont lieu les sauts. Dans la théorie des variables aléatoires indépendantes ces schémas donnent lieu à des lois de probabilité généralisant la loi de Poisson (cf. p. e. Khintchine) ¹ Ergebnisse der Mathematik, Asympt. Gesetze, 1933. . Ils se présentent très naturellement dans le cas où le nombre des positions que le point mobile peut occuper est fini, et les distributions de probabilité qu'on rencontre ainsi ont été étudiées par MM. Kolmogoroff ² Math. Ann. t. 104, pp. 415–58, 1931. , Fréchet ³ voir p. e. Recherches Théoriques Modernes, livre I, pp. 231–54, Gauthiers-Villars, 1938. et Hostinský. ⁴ cf. p. e. Journal de Math., t. 16, pp. 267–84, 1937. On comprend très aisément que dans un mouvement où il n'y a qu'un nombre fini de déplacements il n'y a pas lieu d'attacher de l'importance à la structure topologique de l'espace dans lequel a lieu le mouvement et, effectivement, M. Pospísil ⁵ ?asopis Mat. Fys., t. 65, p. 64–76, 1936. a, en généralisant les méthodes de M. B. Hostinsktý, étudié des mouvements ayant lieu dans des espaces abstraits quelconques, mais dans cette étude, on est obligé d'opérer avec des fonctions d'ensembles abstraits, des corps d'ensembles etc., ce qui nuit un peu au caractère intuitif du problème.     

A cross-currency Lévy market model

The Lévy Libor or market model which was introduced in Eberlein and Özkan (The Lévy Libor model. Financ. Stochast., 2005, 9, 327–348) is extended to a multi-currency setting. As an application we derive closed form pricing formulas for cross-currency derivatives. Foreign caps and floors and cross-currency swaps are studied in detail. Numerically efficient pricing algorithms based on bilateral Laplace transforms are derived. A calibration example is given for a two-currency setting (EUR, USD).     

Sur le calcul numérique des moments ordinaires et des moments composés d'une distribution statistique

Abstract

Considérons une distribution statistique (empirique ou théorique) x _v (v = 1, 2 ... n) désignant les valeurs que peut assumer une variable fortuite une-dimensionelle x, et P_v (v= 1,2 ... n) désignant les fréquences observées (absolues ou relatives) ou bien les probabilités des valeurs x_v .     

On an analogue of Cramér-Rao's inequality

Abstract

Rao [1] and simultaneously Cramér [2, 3] have shown that if f (x, θ) is the probability density function of a distribution involving an unknown parameter θ and distributed over the range α ? x ? b, where a and b are independent of θ, and if x ₁ x ₂ ... x _n is a random sample of n independent observations from this distribution, the variance of any estimate unbiased for Ψ (θ), satisfies the inequality where E denotes mathematical expectation and is Fisher's information index about θ. In (1), equality holds if, and only if, θ* is sufficient for θ. This inequality is further generalized to the multi-parametric case.     

Symmetry and duality in Lévy markets

The aim of this paper is to introduce the notion of symmetry in a Lévy market. This notion appears as a particular case of a general known relation between prices of put and call options, of both the European and the American type, which is also reviewed in the paper, and that we call put–call duality. Symmetric Lévy markets have the distinctive feature of producing symmetric smile curves, in the log of strike/futures prices.

Put–call duality is obtained as a consequence of a change of the risk neutral probability measure through Girsanov's theorem, when considering the discounted and reinvested stock price as the numeraire. Symmetry is defined when a certain law before and after the change of measure through Girsanov's theorem coincides. A parameter characterizing the departure from symmetry is introduced, and a necessary and sufficient condition for symmetry to hold is obtained, in terms of the jump measure of the Lévy process, answering a question raised by Carr and Chesney (American put call symmetry, preprint, 1996 Carr, P and Chesney, M. 1996. American put call symmetry. preprint [Google Scholar]). Some empirical evidence is shown, supporting that, in general, markets are not symmetric.     

Variance swaps on time-changed Lévy processes

We prove that a multiple of a log contract prices a variance swap, under arbitrary exponential Lévy dynamics, stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the driving Lévy process, subject to integrability conditions. We solve for the multiplier, which depends only on the Lévy process, not on the clock. In the case of an arbitrary continuous underlying returns process, the multiplier is 2, which recovers the standard no-jump variance swap pricing formula. In the presence of negatively skewed jump risk, however, we prove that the multiplier exceeds 2, which agrees with calibrations of time-changed Lévy processes to equity options data. Moreover, we show that discrete sampling increases variance swap values, under an independence condition; so if the commonly quoted multiple 2 undervalues the continuously sampled variance, then it undervalues even more the discretely sampled variance. Our valuations admit enforcement, in some cases, by hedging strategies which perfectly replicate variance swaps by holding log contracts and trading the underlying.     

Spectral calibration of exponential Lévy models

We investigate the problem of calibrating an exponential Lévy model based on market prices of vanilla options. We show that this inverse problem is in general severely ill-posed and we derive exact minimax rates of convergence. The estimation procedure we propose is based on the explicit inversion of the option price formula in the spectral domain and a cut-off scheme for high frequencies as regularisation.     

Sur la notion de prime d'épargne

Abstract

Il est bien connu que la réserve mathématique W (t) d'une assurance très générale sur la vie satisfait à une équation différentielle de Thiele, à savoir où l'on a

• δ (t) = le taux instantané ou l'intensité d'intérêt,

• μ (t) = l'intensité de mourir,

• P (t = la prime par unité du temps,

• S (t) = le capital assuré en cas de décès.

     

Lévy processes driven by stochastic volatility

In this paper we extend option pricing under Lévy dynamics, by assuming that the volatility of the Lévy process is stochastic. We, therefore, develop the analog of the standard stochastic volatility models, when the underlying process is not a standard (unit variance) Brownian motion, but rather a standardized Lévy process. We present a methodology that allows one to compute option prices, under virtually any set of diffusive dynamics for the parameters of the volatility process. First, we use ‘local consistency’ arguments to approximate the volatility process with a finite, but sufficiently dense Markov chain; we then use this regime switching approximation to efficiently compute option prices using Fourier inversion. A detailed example, based on a generalization of the popular stochastic volatility model of Heston (Rev Financial Stud 6 (1993) 327), is used to illustrate the implementation of the algorithms. Computer code is available at www.theponytail.net/