Eric Jondeau

Conditional volatility, skewness, and kurtosis: existence, persistence, and comovements

Recent portfolio-choice, asset-pricing, value-at-risk, and option-valuation models highlight the importance of modeling the asymmetry and tail-fatness of returns. These characteristics are captured by the skewness and the kurtosis. We characterize the maximal range of skewness and kurtosis for which a density exists and show that the generalized Student-t distribution spans a large domain in the maximal set. We use this distribution to model innovations of a GARCH type model, where parameters are conditional. After demonstrating that an autoregressive speci5cation of the parameters may yield spurious results, we estimate and test restrictions of the model, for a set of daily stock-index and foreign-exchange returns. The estimation is implemented as a constrained optimization via a sequential quadratic programming algorithm. Adequacy tests demonstrate the importance of a time-varying distribution for the innovations. In almost all series, we 5nd time dependency of the asymmetry parameter, whereas the degree-of-freedom parameter is generally found to be constant over time. We also provide evidence that skewness is strongly persistent, but kurtosis is much less so. A simulation validates our estimations and we conjecture that normality holds for the estimates. In a cross-section setting, we also document covariability of moments beyond volatility, suggesting that extreme realizations tend to occur simultaneously on di9erent markets. ? 2002 Elsevier Science B.V. All rights reserved.

Optimal Portfolio Allocation under Higher Moments

We evaluate how departure from normality may affect the allocation of assets. A Taylor series expansion of the expected utility allows to focus on certain moments and to compute numerically the optimal portfolio allocation. A decisive advantage of this approach is that it remains operational even if a large number of assets is involved. We show that under moderate non-normality the mean-variance criterion provides a good approximation of the expected utility maximization. In contrast, under large departure from normality (as found in some stocks in mature markets or in some stock indices in emerging markets), the mean-variance criterion may fail to approximate the expected utility correctly. In such cases, the three-moment or four-moment optimization strategies may provide a good approximation of the expected utility.

Entropy densities with an application to autoregressive conditional skewness and kurtosis

The entropy principle yields, for a given set of moments, a density that involves the smallest amount of prior information,. We first show how entropy densities may be constructed in a numerically efficient way as the minimization of a potential. Next, for the case where the first four moments are given, we characterize the skewness-kurtosis domain for which densities are defined. This domain is found to be much larger than for Hermite or Edgeworth expansions. Last, we show how this technique can be used to estimate a GARCH model where skewness and kurtosis are time varying.

Gram-Charlier densities

Abstract The Gram–Charlier expansion, where skewness and kurtosis directly appear as parameters, has become popular in Finance as a generalization of the normal density. We show how positivity constraints can be numerically implemented, thereby guaranteeing that the expansion defines a density. The constrained expansion can be referred to as a Gram–Charlier density. First, we apply our method to the estimation of risk neutral densities. Then, we assess the statistical properties of maximum-likelihood estimates of Gram–Charlier densities. Lastly, we apply the framework to the estimation of a GARCH model where the conditional density is a Gram–Charlier density.

Does Correlation between Stock Returns Really Increase during Turbulent Period

Correlations between international equity markets are often claimed to increase during periods of high volatility, therefore the benefits of international diversification are reduced when they are most needed, i.e. during crises. In this paper, we investigate the relationship between international correlation and stock-market turbulence. We estimate a multivariate Markov-switching model, in which the correlation matrix is allowed to vary across regimes. Subsequently, we test the null hypothesis that correlations are regime independent. Using weekly stock returns for the S&P, the DAX and the FTSE over the period 1988-1999, we find that international correlations significantly increased during turbulent periods.

Long-Run Causality, with an Application to International Links between Long-Term Interest Rates

In this paper we give a precise definition of long-run causality in a multivariate non-stationary, possibly cointegrated, framework. A variable is said to be causal for another in the long-run if knowledge of the past of the former improves long-run predictions of the latter. In a VAR framework, we show that long-run non-causality can be easily tested with a Wald statistics, conditionally on the cointegration rank. The methodology is used to study long-run causal links between US, German, and French long-term interest rates from January 1990 to June 1997. Copyright 1999 by Blackwell Publishing Ltd

Testing for differences in the tails of stock-market returns

In this paper, we use a database consisting of daily stock-returns for 20 countries to test for similarities between the left and right tail of returns as well as for cross-sectional differences. To mitigate the issue of dependency between stock returns, we estimate the distribution of extremes over subsamples of two months. We document a good fit of the model and show that the left and right tails of returns behave very similarly. Across countries, we find that extremes are located at different levels and that their dispersion varies. On the other hand, the tail index, characterizing large extreme realizations is found to be constant worldwide. Our results are not due to a lack of power. We also discuss the results from an economic point of view.

Reading Interest Rate and Bond Futures Options' Smiles: How PIBOR and National Operators Appreciated the 1997 French Snap Election

The aim of this paper is to compare various methods which extract a Risk Neutral Density (RND) out of PIBOR as well as of Notional interest rate futures options and to investigate how traders reacted to a political event. We first focus on 5 dates surrounding the 1997 snap election and several methods: Black (1976), a mixture of lognormals (as in Melick and Thomas, 1997), an Hermite expansion (as in Abken, Madan, and Ramamurtie, 1996), and a method based on Maximum Entropy (following Kelly and Buchen, 1996). By and large the various methods give similar RNDs. Yet, the Hermite expansion approach, by allowing for somewhat dirty options prices, by providing a good fit to options prices, and by being very fast is the retained method for the data at hand. We then consider a daily panel of options running from February 1997 to July 1997. After constructing standardized options, i.e. with a fixed time to maturity, we find that operators in both markets anticipated the snap election a few days before the official announcement and that a substantial amount of political uncertainty subsisted even a month after the elections. The greater liquidity of PIBOR options eases information extraction.

Reading the Smile: The Message Conveyed by Methods which Infer Risk Neutral Densities

In this study we compare the quality and information content of risk neutral densities obtained by various methods. We consider a non-parametric method based on a mixture of log-normal densities, the semi-parametric ones based on an Hermite approximation of Madan and Milne, or based on an Edgeworth expansion of Jarrow and Rudd, the parametric approach of Malz which assumes a jump-diffusion for the underlying process, and eventually Hestons approach assuming a stochastic volatility model. We apply those models on FRF/DEM exchange rate options for two dates, for various maturities. Models differ when important news hit the market (here the 1997 snap elections). The non-parametric model provides a good fit to options prices but is unable under critical circumstances to provide as much information about market participants expectations than Malzs jump-diffusion model.

Conditional Volatility, Skewness, and Kurtosis: Existence and Persistence

Recent portfolio choice, asset pricing, and option valuation models highlight the importance of skewness and kurtosis. Since skewness and kurtosis are related to extreme variations, they are also important for Value-at-Risk measurements. Our framework builds on a GARCH model with a conditional generalized-t distribution for residuals. We compute the skewness and kurtosis for this model and compare the range of these moments with the maximal theoretical moments. Our model, thus allows for time-varying conditional skewness and kurtosis. We implement the model as a constrained optimization with possibly several thousand restrictions on the dynamics. A sequential quadratic programming algorithm successfully estimates all the models, on a PC, within at most 50 seconds. Estimators, obtained with logistically-constrained dynamics, have different properties. We apply this model to daily and weekly foreign exchange returns, stock returns, and interest-rate changes. This finding is consistent with findings from extreme value theory. Kurtosis exists on fewer dates and for fewer series. There is little evidence, at the weekly frequency, of time-variability of conditional higher moments. Transition matrices document that agitated stares come as a surprise and that there is a certain persistence in moments beyond volatility. For exchange-rate and stock-market data, cross-sectionally and at daily frequency, we also document co-variability of moments beyond volatility.