What Data Should Be Used to Price Options?

In this paper we propose a generic procedure for estimating and pricing options in the context of stochastic volatility models using simultaneously the fundamental price and a set of option contracts. We appraise univariate and multivariate estimation of the model in terms of pricing and hedging performance. Our results, based on the S&P 500 index contract, show that the univariate approach only involving options by and large dominates. A by-product of this finding is that we uncover a remarkably simple volatility extraction filter based on a polynomial lag structure of implied volatilities. The bivariate approach involving both the fundamental and an option appears useful when the information from the cash market provides support via the conditional kurtosis to price options. This is the case for some long-term options. Moreover, having estimated separately the risk-neutral and objective measures allows us to appraise the typical risk-neutral representations used in the literature. Using Heston's (1993) model as example we show that the usual transformation from objective to risk neutral density is not supported by the data. Nous présentons une procédure générique pour l'estimation et l'évaluation de modèles d'options avec volatilité stochastique où le sousjacent et un ensemble de contrats d'options sont utilisés simultanément. Nos résultats démontrent qu'un modèle univarié avec seulement des données d'options domine en terme d'erreurs de prix hors-échantillon et en terme de couverture. Nous trouvons également un filtre d'extraction pour la volatilité latente qui est basé sur un polynome de retards de volatilités implicites. Ayant simultanément la probabilité de risque neutre et la probabilité objective, nous pouvons vérifier, dans le contexte du modèle de Heston, si la transformation usuelle est empiriquement plausible. Nous rejetons le changement de mesure supposé dans ce modèle.Derivative securities, efficient method of moments, state price densities, stochastic volatility models, filtering, Titres dérivés, méthode de moments efficaces, prix d'états, filtrage, volatilité stochastique

A New Class of Stochastic Volatility Models with Jumps: Theory and Estimation

The purpose of this paper is to propose a new class of jump diffusions which feature both stochastic volatility and random intensity jumps. Previous studies have focussed primarily on pure jump processes with constant intensity and log-normal jumps or constant jump intensity combined with a one factor stochastic volatility model. We introduce several generalizations which can better accommodate several empirical features of returns data. In their most general form we introduce a class of processes which nests jump-diffusions previously considered in empirical work and includes the affine class of random intensity models studied by Bates (1998) and Duffie, Pan and Singleton (1998) but also allows for non-affine random intensity jump components. We attain the generality of our specification through a generic Lévy process characterization of the jump component. The processes we introduce share the desirable feature with the affine class that they yield analytically tractable and explicit option pricing formula. The non-affine class of processes we study include specifications where the random intensity jump component depends on the size of the previous jump which represent an alternative to affine random intensity jump processes which feature correlation between the stochastic volatility and jump component. We also allow for and experiment with different empirical specifications of the jump size distributions. We use two types of data sets. One involves the S&P500 and the other comprises of 100 years of daily Dow Jones index. The former is a return series often used in the literature and allows us to compare our results with previous studies. The latter has the advantage to provide a long time series and enhances the possibility of estimating the jump component more precisely. The non-affine random intensity jump processes are more parsimonious than the affine class and appear to fit the data much better. Nous présentons une nouvelle classe de processus à sauts avec volatilité stochastique. Cette nouvelle classe généralise les modèles affinés proposés par Duffie, Pan et Singleton (1998). La généralité se manifeste par une représentation générique des sauts par un processus de Lévy. La classe des processus que nous présentons nous fournit également des prix d'options. Une application empirique démontre la présence de sauts dans des séries financières telles le S&P500 et le Dow Jones. De plus, les processus n'ont pas une intensité constante. Nous analysons plusieurs spécifications empiriques.Efficient method of moments, Poisson processes, jump processes, stochastic volatility models, filtering, Processus à sauts, mesures de Lévy, modèles à volatilité stochastique

Efficient Estimation of Jump Diffusions and General Dynamic Models with a Continuum of Moment Conditions

A general estimation approach combining the attractive features of method of moments with the efficiency of ML is proposed. The moment conditions are computed via the characteristic function. The two major difficulties with the implementation is that one needs to use an infinite set of moment conditions leading to the singularity of the covariance matrix in the GMM context, and the optimal instrument yielding the ML efficiency was previously shown to depend on the unknown probability density function. We resolve the two problems simultaneously in the framework of C-GMM (GMM with a continuum of moment conditions). First, we prove asymptotic properties of the C-GMM estimator applied to dependent data and then provide a reformulation of the estimator that enhances its computational ease. Second, we propose to span the unknown optimal instrument by an infinite basis consisting of simple exponential functions. Since the estimation framework already relies on a continuum of moment conditions, adding a continuum of spanning functions does not pose any problems. As a result, we achieve ML efficiency when we use the values of conditional CF indexed by its argument as moment functions. We also introduce HAC-type estimators so that the estimation methods are not restricted to settings involving martingale difference sequences. Hence, our methods apply to Markovian and nail-Markovian dynamic models. Finally, a simulated method of moments type estimator is proposed to deal with the cases where the characteristic function does not have a closed-form expression. Extensive Monte-Carlo study based on the models typically used in term-structure literature favorably documents the performance of our methodology. L'estimation des processus de diffusion (affine ou à sauts) est problématique car l'expression de la vraisemblance n'est pas disponible. D'un autre côté, la fonction caractéristique de ces modèles est souvent connue. Cet article propose un estimateur du type méthode des moments généralisés (GMM) fondé sur la fonction caractéristique. Comme l'on dispose d'un continuum de conditions de moments, on utilise une méthode spécifique appelée C-GMM. On dérive les propriétés asymptotiques de l'estimateur et discute son implémentation en pratique. Dans le contexte d'un processus markovien, une condition de moment conditionnelle résulte de la fonction caractéristique conditionnelle. Une question importante est le choix de l'instrument optimal. On montre que, lorsque l'instrument est une fonction exponentielle, l'estimateur C-GMM est asymptotiquement aussi efficace que l'estimateur du maximum de vraisemblance. Il faut noter que la méthode C-GMM n'est pas limitée aux processus markoviens et s'applique à des modèles dynamiques très généraux. De plus, on propose une méthode des moments simulés qui permet de traiter le cas où l'expression de la fonction caractéristique n'est pas connue. Finalement, une étude de Monte Carlo sur des modèles fréquemment utilisés en finance montre que notre estimateur a de bonnes propriétés.Asymptotic efficiency. Characteristic function. GMM. Diffusion processes. Simulated Method of Moments, Efficacité asymptotique. Fonction caractéristique. Méthode des moments généralisés. Méthode des moments simulés. Processus de diffusion

Disasters implied by equity index options

We use prices of equity index options to quantify the impact of extreme events on asset returns. We define extreme events as departures from normality of the log of the pricing kernel and summarize their impact with high-order cumulants: skewness, kurtosis, and so on. We show that high-order cumulants are quantitatively important in both representative-agent models with disasters and in a statistical pricing model estimated from equity index options. Option prices thus provide independent confirmation of the impact of extreme events on asset returns, but they imply a more modest distribution of them.

Sources of Entropy in Representative Agent Models

We propose two metrics for asset pricing models and apply them to representative agent models with recursive preferences, habits, and jumps. The metrics describe the pricing kernel’s dispersion (the entropy of the title) and dynamics (time dependence, a measure of how entropy varies over different time horizons). We show how each model generates entropy and time dependence and compare their magnitudes to estimates derived from asset returns. This exercise — and transparent loglinear approximations — clarifies the mechanisms underlying these models. It also reveals, in some cases, tension between entropy, which should be large enough to account for observed excess returns, and time dependence, which should be small enough to account for mean yield spreads.

Alternative Models for Stock Price Dynamics

This paper evaluates the role of various volatility specifications, such as multiple stochastic volatility (SV) factors and jump components, in appropriate modeling of equity return distributions. We use estimation technology that facilitates non-nested model comparisons and use a long data set which provides rich information about the conditional and unconditional distribution of returns. We consider two broad families of models: (1) the multifactor loglinear family, and (2) the affine-jump family. Both classes of models have attracted much attention in the derivatives and econometrics literatures. There are various trade-offs in considering such diverse specifications. If pure diffusion SV models are chosen over jump diffusions, it has important implications for hedging strategies. If logaritmic models are chosen over affine ones, it may seriously complicate option pricing. Comparing many different specifications of pure diffusion multi-factor models and jump diffusion models, we find that (1) log linear models have to be extented to 2 factors with feedback in the mean reverting factor, (2) affine models have to have a jumps in returns, stochastic volatility and probably both. Models (1) and (2) are observationally equivalent on the data set in hand. In either (1) or (2) the key is that the volatility can move violently. As we obtain models with comparable empirical fit, one must make a choice based on arguments other than statistical goodness of fit criteria. The considerations include facility to price options, to hedge and parsimony. The affine specification with jumps in volatility might therefore be preferred because of the closed-form derivatives prices. Nous examinons un ensemble de diffusions avec volatilité stochastique et de sauts afin de modéliser la distribution des rendements d'actifs boursiers. Puisque certains modèles sont non-emboîtés, nous utilisons la méthode EMM afin d'étudier et de comparer le comportement des différents modèles.Efficient method of moments, Poisson jump processes, stochastic volatility models, Processus de diffusions, processus Poisson, volatilité stochastique

Microstructure and mechanical properties of V–Cr–Zr alloy with carbide and oxide strengthening

A comparative study of the effectiveness of carbide and oxide types of strengthening of V–Cr–Zr alloy was carried out by means of a comprehensive certification of structural-phase state parameters and measuring the mechanical properties characteristics. It has been shown that the use of chemical-heat treatment contributes to a significant increase in the thermal stability of the microstructure and mechanical properties of V–Cr–Zr alloy in comparison with carbide strengthening under the conditions of thermomechanical treatment. A controlled increase in the volume fraction of fine particles based on ZrO2, along with an increase in the concentration of oxygen in the solid solution, leads to a decrease in the rate of oxides coagulation and an increase in the thermal stability of high disperse heterophase structure. These effects contribute to the retention of high defect structural states with nonzero values of crystal lattice curvature even after high-temperature (0.67 Tmelt) anneals. The high efficiency of dispersion and substructural strengthening is a consequence of blocking dislocation slip by fine particles stabilized by oxygen in a solid solution

Microstructure and mechanical properties of vanadium alloys after thermomechanical treatments

The results of investigation of dispersion strengthening effect on parameters of structural-phase states and characteristics of short-term strength and ductility of vanadium alloys of V–4Ti–4Cr, V–2.4Zr–0.25C, V–1.2Zr–8.8Cr and V–1.7Zr–4.2Cr–7.6W systems with different concentration of interstitial elements after optimized thermomechanical treatment mode were summarized. It was shown that for effective realization of dispersion strengthening by Orowan-type mechanism at least 25–50% of the initial volume fraction of coarse particles should be transformed into fine-disperse state and redistributed over the volume of material