Nonparametric estimation of the leverage effect: a trade-off between robustness and efficiency

We consider two new approaches to nonparametric estimation of the leverage effect. The first approach uses stock prices alone. The second approach uses the data on stock prices as well as a certain volatility instrument, such as the CBOE volatility index (VIX) or the Black-Scholes implied volatility. The theoretical justification for the instrument-based estimator relies on a certain invariance property, which can be exploited when high frequency data is available. The price-only estimator is more robust since it is valid under weaker assumptions. However, in the presence of a valid volatility instrument, the price-only estimator is inefficient as the instrument-based estimator has a faster rate of convergence. We consider two empirical applications, in which we study the relationship between the leverage effect and the debt-to-equity ratio, credit risk, and illiquidity.

The leverage effect in the UK stock market

This study seeks to explain the leverage in UK stock returns by reference to the return volatility, leverage and size characteristics of UK companies. A leverage effect is found that is stronger for smaller companies and has greater explanatory power over the returns of smaller companies. The properties of a theoretical model that predicts that companies with higher leverage ratios will experience greater leverage effects are explored. On examining leverage ratio data, it is found that there is a propensity for smaller companies to have higher leverage ratios. The transmission of volatility shocks between the companies is also examined and it is found that the volatility of larger firm returns is important in determining both the volatility and returns of smaller firms, but not the reverse. Moreover, it is found that where volatility spillovers are important, they improve out-of-sample volatility forecasts. © 2005 Taylor & Francis Group Ltd.

Volume, volatility, and leverage: A dynamic analysis

This paper uses dynamic impulse response analysis to investigate the interrelationships among stock price volatility, trading volume, and the leverage effect. Dynamic impulse response analysis is a technique for analyzing the multi-step-ahead characteristics of a nonparametric estimate of the one-step conditional density of a strictly stationary process. The technique is the generalization to a nonlinear process of Sims-style impulse response analysis for linear models. In this paper, we refine the technique and apply it to a long panel of daily observations on the price and trading volume of four stocks actively traded on the NYSE: Boeing, Coca-Cola, IBM, and MMM.

Asymmetric Smiles, Leverage Effects and Structural Parameters.

In this paper, we characterize the asymmetries of the smile through multiple leverage effects in a stochastic dynamic asset pricing framework. The dependence between price movements and future volatility is introduced through a set of latent state variables. These latent variables can capture not only the volatility risk and the interest rate risk which potentially affect option prices, but also any kind of correlation risk and jump risk. The standard financial leverage effect is produced by a cross-correlation effect between the state variables which enter into the stochastic volatility process of the stock price and the stock price process itself. However, we provide a more general framework where asymmetric implied volatility curves result from any source of instantaneous correlation between the state variables and either the return on the stock or the stochastic discount factor. In order to draw the shapes of the implied volatility curves generated by a model with latent variables, we specify an equilibrium-based stochastic discount factor with time non-separable preferences. When we calibrate this model to empirically reasonable values of the parameters, we are able to reproduce the various types of implied volatility curves inferred from option market data.

Essays on volatility and liquidity in financial markets

The price formation of financial assets is a complex process. It extends beyond the standard economic paradigm of supply and demand to the understanding of the dynamic behavior of price variability, the price impact of information, and the implications of trading behavior of market participants on prices. In this thesis, I study aggregate market and individual assets volatility, liquidity dimensions, and causes of mispricing for US equities over a recent sample period. How volatility forecasts are modeled, what determines intradaily jumps and causes changes in intradaily volatility and what drives the premium of traded equity indexes? Are they induced, for example, by the information content of lagged volatility and return parameters or by macroeconomic news, changes in liquidity and volatility? Besides satisfying our intellectual curiosity, answers to these questions are of direct importance to investors developing trading strategies, policy makers evaluating macroeconomic policies and to arbitrageurs exploiting mispricing in exchange-traded funds. Results show that the leverage effect and lagged absolute returns improve forecasts of continuous components of daily realized volatility as well as jumps. Implied volatility does not subsume the information content of lagged returns in forecasting realized volatility and its components. The reported results are linked to the heterogeneous market hypothesis and demonstrate the validity of extending the hypothesis to returns. Depth shocks, signed order flow, the number of trades, and resiliency are the most important determinants of intradaily volatility. In contrast, spread shock and resiliency are predictive of signed intradaily jumps. There are fewer macroeconomic news announcement surprises that cause extreme price movements or jumps than those that elevate intradaily volatility. Finally, the premium of exchange-traded funds is significantly associated with momentum in net asset value and a number of liquidity parameters including the spread, traded volume, and illiquidity. The mispricing of industry exchange traded funds suggest that limits to arbitrage are driven by potential illiquidity.

Do benchmark African equity indices exhibit the stylized facts?

This paper investigates if benchmark African equity indices exhibit the stylized facts reported for financial time-series returns. The returns distributions of the Africa All-Share, Large, Medium and Small Company Indices were found to be leptokurtotic, had fat-tails, over time experienced volatility clustering and exhibited long memory in volatility. Both the All-Share and Large Company Indices were found to exhibit leverage effects. In contrast, positive shocks had a greater impact on future volatility for the Small Company Index which implies a reverse leverage effect. This finding could reflect a bull/bubble market for small capitalisation stocks in Africa.

A Theoretical Comparison Between Integrated and Realized Volatilies

In this paper, we provide both qualitative and quantitative measures of the cost of measuring the integrated volatility by the realized volatility when the frequency of observation is fixed. We start by characterizing for a general diffusion the difference between the realized and the integrated volatilities for a given frequency of observations. Then, we compute the mean and variance of this noise and the correlation between the noise and the integrated volatility in the Eigenfunction Stochastic Volatility model of Meddahi (2001a). This model has, as special examples, log-normal, affine, and GARCH diffusion models. Using some previous empirical works, we show that the standard deviation of the noise is not negligible with respect to the mean and the standard deviation of the integrated volatility, even if one considers returns at five minutes. We also propose a simple approach to capture the information about the integrated volatility contained in the returns through the leverage effect.

Risk Aversion, Intertemporal Substitution, and Option Pricing

This paper develops a general stochastic framework and an equilibrium asset pricing model that make clear how attitudes towards intertemporal substitution and risk matter for option pricing. In particular, we show under which statistical conditions option pricing formulas are not preference-free, in other words, when preferences are not hidden in the stock and bond prices as they are in the standard Black and Scholes (BS) or Hull and White (HW) pricing formulas. The dependence of option prices on preference parameters comes from several instantaneous causality effects such as the so-called leverage effect. We also emphasize that the most standard asset pricing models (CAPM for the stock and BS or HW preference-free option pricing) are valid under the same stochastic setting (typically the absence of leverage effect), regardless of preference parameter values. Even though we propose a general non-preference-free option pricing formula, we always keep in mind that the BS formula is dominant both as a theoretical reference model and as a tool for practitioners. Another contribution of the paper is to characterize why the BS formula is such a benchmark. We show that, as soon as we are ready to accept a basic property of option prices, namely their homogeneity of degree one with respect to the pair formed by the underlying stock price and the strike price, the necessary statistical hypotheses for homogeneity provide BS-shaped option prices in equilibrium. This BS-shaped option-pricing formula allows us to derive interesting characterizations of the volatility smile, that is, the pattern of BS implicit volatilities as a function of the option moneyness. First, the asymmetry of the smile is shown to be equivalent to a particular form of asymmetry of the equivalent martingale measure. Second, this asymmetry appears precisely when there is either a premium on an instantaneous interest rate risk or on a generalized leverage effect or both, in other words, whenever the option pricing formula is not preference-free. Therefore, the main conclusion of our analysis for practitioners should be that an asymmetric smile is indicative of the relevance of preference parameters to price options.

Aggregations and Marginalization of GARCH and Stochastic Volatility Models

The GARCH and Stochastic Volatility paradigms are often brought into conflict as two competitive views of the appropriate conditional variance concept : conditional variance given past values of the same series or conditional variance given a larger past information (including possibly unobservable state variables). The main thesis of this paper is that, since in general the econometrician has no idea about something like a structural level of disaggregation, a well-written volatility model should be specified in such a way that one is always allowed to reduce the information set without invalidating the model. To this respect, the debate between observable past information (in the GARCH spirit) versus unobservable conditioning information (in the state-space spirit) is irrelevant. In this paper, we stress a square-root autoregressive stochastic volatility (SR-SARV) model which remains true to the GARCH paradigm of ARMA dynamics for squared innovations but weakens the GARCH structure in order to obtain required robustness properties with respect to various kinds of aggregation. It is shown that the lack of robustness of the usual GARCH setting is due to two very restrictive assumptions : perfect linear correlation between squared innovations and conditional variance on the one hand and linear relationship between the conditional variance of the future conditional variance and the squared conditional variance on the other hand. By relaxing these assumptions, thanks to a state-space setting, we obtain aggregation results without renouncing to the conditional variance concept (and related leverage effects), as it is the case for the recently suggested weak GARCH model which gets aggregation results by replacing conditional expectations by linear projections on symmetric past innovations. Moreover, unlike the weak GARCH literature, we are able to define multivariate models, including higher order dynamics and risk premiums (in the spirit of GARCH (p,p) and GARCH in mean) and to derive conditional moment restrictions well suited for statistical inference. Finally, we are able to characterize the exact relationships between our SR-SARV models (including higher order dynamics, leverage effect and in-mean effect), usual GARCH models and continuous time stochastic volatility models, so that previous results about aggregation of weak GARCH and continuous time GARCH modeling can be recovered in our framework.

The Shape of the Risk Premium: Evidence from a Semiparametric Garch Model.

We examine the relationship between the risk premium on the S&P 500 index return and its conditional variance. We use the SMEGARCH - Semiparametric-Mean EGARCH - model in which the conditional variance process is EGARCH while the conditional mean is an arbitrary function of the conditional variance. For monthly S&P 500 excess returns, the relationship between the two moments that we uncover is nonlinear and nonmonotonic. Moreover, we find considerable persistence in the conditional variance as well as a leverage effect, as documented by others. Moreover, the shape of these relationships seems to be relatively stable over time.

ARMA Representation of Integrated and Realized Variances

This paper derives the ARMA representation of integrated and realized variances when the spot variance depends linearly on two autoregressive factors, i.e., SR SARV(2) models. This class of processes includes affine, GARCH diffusion, CEV models, as well as the eigenfunction stochastic volatility and the positive Ornstein-Uhlenbeck models. We also study the leverage effect case, the relationship between weak GARCH representation of returns and the ARMA representation of realized variances. Finally, various empirical implications of these ARMA representations are considered. We find that it is possible that some parameters of the ARMA representation are negative. Hence, the positiveness of the expected values of integrated or realized variances is not guaranteed. We also find that for some frequencies of observations, the continuous time model parameters may be weakly or not identified through the ARMA representation of realized variances.

Changements démographiques et inégalités éducatives à Ouagadougou

Depuis plusieurs décennies, des études empiriques réalisées sur plusieurs pays développés ou en émergence ont montré que la baisse de la taille de la famille favorise l’investissement dans l’éducation des enfants, expliquant qu’un nombre élevé d’enfants a un effet d’amenuisement des ressources familiales. Les retombées positives de la baisse de la fécondité sur l’éducation sont largement étudiées et connues. En dépit des résultats controversés des premières études portant sur les pays de l’Afrique de l’Ouest, les récentes études empiriques tendent à confirmer l’effet positif de la baisse de la taille de la famille dans le contexte africain, du moins en milieu urbain. Par contre, jusqu’à présent, très peu d’études semblent intéressées à analyser la répartition de ces retombées entre les enfants, et encore moins à comprendre comment ces dernières affecteraient la structure des inégalités éducatives existantes. Notre étude s’intéresse à explorer la potentielle dimension démographique des inégalités socioéconomiques, notamment les inégalités éducatives dans le contexte de la baisse de la fécondité. Elle vise à apporter des évidences empiriques sur le lien entre la réduction de la taille de la famille et les inégalités éducatives au sein des ménages dans le contexte d’Ouagadougou, Capitale du Burkina Faso, qui connait depuis quelques décennies la chute de la fécondité. Elle analyse aussi l’effet de cette réduction sur la transmission intergénérationnelle des désavantages éducatifs. Pour ce faire, nous proposons un cadre conceptuel pour comprendre les mécanismes par lesquels la relation entre la réduction de la taille de la famille et les inégalités éducatives se tisse. Ce cadre conceptuel s’appuie sur une recension des écrits de divers auteurs à ce sujet. Par la suite, nous procédons à des analyses empiriques permettant de tester ces liens en utilisant les données du projet Demtrend collectées. Les résultats empiriques sont présentés sous forme d’articles scientifiques. Les conclusions du premier article indiquent que la relation entre le nombre d’enfants de la famille et l’éducation varie selon le contexte socioéconomique. En effet, pour les générations qui ont grandi dans un contexte socioéconomique colonial et postcolonial, où le mode de production était essentiellement agricole et l’éducation formelle n’était pas encore valorisée sur le marché du travail, la relation est très faible et positive. Par contre, pour les récentes générations, nous avons observé que la relation devient négative et fortement significative. De plus, les résultats de cet article suggèrent aussi que la famille d’origine des femmes a une incidence significative sur leur comportement de fécondité. Les femmes dont la mère avait un niveau de scolarité élevé (et étaient de statut socioéconomique aisé) ont moins d’enfants comparativement à celles dont leurs parents avaient un faible niveau de scolarité (et pauvres). En retour, leurs enfants sont aussi les plus éduqués. Ce qui sous-tend à un éventuel effet de levier de la réduction de la taille de la famille dans le processus de transmission intergénérationnelle des désavantages éducatifs. Le second article fait une comparaison entre les ménages de grande taille et ceux de petite taille en matière d’inégalités éducatives entre les enfants au sein des ménages familiaux, en considérant le sexe, l’ordre de naissance et les termes d’interaction entre ces deux variables. Les résultats de cet article montrent que généralement les enfants des familles de petite taille sont plus scolarisés et atteignent un niveau d’éducation plus élevé que ceux des grandes familles. Toutefois, les filles ainées des petites familles s’avèrent moins éduquées que leurs pairs. Ce déficit persiste après avoir considéré seulement les ménages familiaux monogames ou encore après le contrôle de la composition de la fratrie. L’émancipation des femmes sur le marché du travail résultant de la réduction de la taille de la famille et la faible contribution des pères dans les activités domestiques expliqueraient en partie cette situation. Malheureusement, nous n’avons pas pu contrôler l’activité économique des mères dans les analyses. Finalement, dans le cadre du troisième et dernier article, nous avons examiné l’effet d’avoir été confié par le passé sur les inégalités éducatives au sein de la fratrie, en comparant ceux qui ont été confiés aux autres membres de leur fratrie qui n’ont jamais été confiés. Dans cet article, nous avons considéré l’aspect hétérogène du confiage en le différenciant selon le sexe, la relation de la mère avec le chef du ménage d’accueil et l’âge auquel l’enfant a été confié. Les résultats montrent qu’avoir été confié dans le passé influence négativement le parcours scolaire des enfants. Cependant, cet effet négatif reste fort et significatif que pour les filles qui ont été confiées après leurs 10 ans d’âge. Un profil qui correspond à la demande de main-d’œuvre en milieu urbain pour l’accomplissement des tâches domestiques, surtout dans le contexte de la baisse de la taille de la famille et l’émancipation des femmes sur le marché du travail.

Modelando a volatilidade dos retornos de Petrobrás usando dados de alta frequência

O objetivo do presente trabalho é analisar as características empíricas de uma série de retornos de dados em alta freqüência para um dos ativos mais negociados na Bolsa de Valores de São Paulo. Estamos interessados em modelar a volatilidade condicional destes retornos, testando em particular a presença de memória longa, entre outros fenômenos que caracterizam este tipo de dados. Nossa investigação revela que além da memória longa, existe forte sazonalidade intradiária, mas não encontramos evidências de um fato estilizado de retornos de ações, o efeito alavancagem. Utilizamos modelos capazes de captar a memória longa na variância condicional dos retornos dessazonalizados, com resultados superiores a modelos tradicionais de memória curta, com implicações importantes para precificação de opções e de risco de mercado

Estudo e análise estatística no mercado de ações brasileiro

The objective of this paper is to verify and analyze the existence in Brazil of stylized facts observed in financial time series: volatility clustering, probability distributions with fat tails, the presence of long run memory in absolute return time series, absence of linear return autocorrelation, gain/loss asymmetry, aggregative gaussianity, slow absolute return autocorrelation decay, trading volume/volatility correlation and leverage effect. We analyzed intraday prices for 10 stocks traded at the BM&FBovespa, responsible for 52.1% of the Ibovespa portfolio on Sept. 01, 2009. The data analysis confirms the stylized facts, whose behavior is consistent with what is observed in international markets.