this article is to develop new methods for inference and prediction in a simple class of stochastic volatility models in which logarithm of conditional volatility follows an autoregressive (AR) times series model. Unlike the autoregressive conditional heteroscedasticity (ARCH) and gener- alized ARCH (GARCH) models [see Bollerslev, Chou, and Kroner (1992) for a survey of ARCH modeling], both the mean and log-volatility equations have separate error terms. The ease of evaluating the ARCH likelihood function and the ability of the ARCH specification to accommodate the timevarying volatility found in many economic time series has fostered an explosion in the use of ARCH models. On the other hand, the likelihood function for stochastic volatility models is difficult to evaluate, and hence these models have had limited empirical application

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It is widely known that conditional covariances of asset returns change over time.

We exploit the distributional information contained in high-frequency intraday data in constructing a simple conditional moment estimator for stochastic volatility diffusions. The estimator is based on the analytical solutions of the first two conditional moments for the integrated volatility, which is effectively approximated by the quadratic variation of the process. We successfully implement the resulting GMM estimator with high-frequency fiveminute foreign exchange and equity index returns. Our simulation evidence and actual empirical results indicate that the method is very reliable and accurate. The computational speed of the procedure compares very favorably to other existing estimation methods in the literature.

In this paper, we consider temporal aggregation of volatility models. We introduce a semiparametric class of volatility models termed square-root stochastic autoregressive volatility (SR-SARV) and characterized by an autoregressive dynamic of the stochastic variance. Our class encompass the usual GARCH models of Bollerslev (1986), the asymmetric GARCH models of Glosten, Jagannathan and Runkle (1989) and Engle and Ng (1993). Moreover, when the volatility is stochastic, that is there is a second source of randomness, the considered models are characterized by observable multi-period conditional moment restrictions (Hansen, 1985). The SR-SARV class is a natural extension of the weak GARCH models of Drost and Nijman (1993). Our extension has two advantages: i) We allow for asymmetries (skewness, leverage e®ect) that are excluded by the weak GARCH models; ii) we derive observable conditional moment restrictions which are useful for (non linear) inference.

This article develops a direct filtration-based maximum likelihood methodology for estimating the parameters and realizations of latent affine processes. Filtration is conducted in the transform space of characteristic functions, using a version of Bayes ’ rule for recursively updating the joint characteristic function of latent variables and the data conditional upon past data. An application to daily stock market returns over 1953-96 reveals substantial divergences from EMM-based estimates; in particular, more substantial and time-varying jump risk. The implications for pricing stock index options are examined. 3 “The Lion in Affrik and the Bear in Sarmatia are Fierce, but Translated into a Contrary Heaven, are of less Strength and Courage.” Jacob Ziegler; translated by Richard Eden (1555) While models proposing time-varying volatility of asset returns have been around for thirty years, it has proven extraordinarily difficult to estimate the parameters of the underlying volatility process,

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Characterizing asset return dynamics using volatility models is an important part of empirical finance. The existing literature on GARCH models favors some rather complex volatility specifications whose relative performance is usually assessed through their likelihood based on a time-series of asset returns. This paper compares a range of GARCH models along a different dimension, using option prices and returns under the risk-neutral as well as the physical probability measure. We judge the relative performance of various models by evaluating an objective function based on option prices. In contrast with returns-based inference, we find that our option-based objective function favors a relatively parsimonious model. Specifically, when evaluated out-of-sample, our analysis favors a model that besides volatility clustering only allows for a standard leverage effect. JEL Classification: G12

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A volatility model must be able to forecast volatility; this is the central requirement in almost all financial applications. In this paper we outline some stylised facts about volatility that should be incorporated in a model; pronounced persistence and meanreversion, asymmetry such that the sign of an innovation also affects volatility and the possibility of exogenous or pre-determined variables influencing volatility. We use data on the Dow Jones Industrial index to illustrate these stylised facts, and the ability of GARCH-type models to capture these features. We conclude with some challenges for future research in this area. Keywords: volatility modelling, ARCH, GARCH, volatility forecasting. JEL Classification Code : C22 * Please send comments or questions to rengle@stern.nyu.edu. 2 1. INTRODUCTION A volatility model should be able to forecast volatility. Virtually all the financial uses of volatility models entail forecasting aspects of future returns. Typically a volati...