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The Distribution of Realized Exchange Rate Volatility
 Journal of the American Statistical Association
, 2001
"... Using highfrequency data on deutschemark and yen returns against the dollar, we construct modelfree estimates of daily exchange rate volatility and correlation that cover an entire decade. Our estimates, termed realized volatilities and correlations, are not only modelfree, but also approximately ..."
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Using highfrequency data on deutschemark and yen returns against the dollar, we construct modelfree estimates of daily exchange rate volatility and correlation that cover an entire decade. Our estimates, termed realized volatilities and correlations, are not only modelfree, but also approximately free of measurement error under general conditions, which we discuss in detail. Hence, for practical purposes, we may treat the exchange rate volatilities and correlations as observed rather than latent. We do so, and we characterize their joint distribution, both unconditionally and conditionally. Noteworthy results include a simple normalityinducing volatility transformation, high contemporaneous correlation across volatilities, high correlation between correlation and volatilities, pronounced and persistent dynamics in volatilities and correlations, evidence of longmemory dynamics in volatilities and correlations, and remarkably precise scaling laws under temporal aggregation.
Analysis of Stock Market . . .
"... The discrete time ARCH/GARCH model of Engle and Bollarslev has been enormously influential and successful in the modelling of financial data. Recently, Klüppelberg, Lindner, and Maller (2004) introduced the socalled “COGARCH ” model as a continuoustime analogue to the GARCH model. Many aspects of ..."
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The discrete time ARCH/GARCH model of Engle and Bollarslev has been enormously influential and successful in the modelling of financial data. Recently, Klüppelberg, Lindner, and Maller (2004) introduced the socalled “COGARCH ” model as a continuoustime analogue to the GARCH model. Many aspects of the COGARCH have been investigated, including various of its theoretical properties, its relations to other continuoustime models, and the estimation of the parameters in it. We review some of these results in the present paper, and go on to apply the COGARCH to 5minute data on the S&P500 index, in order to illustrate its ability to analyse stochastic volatility in very highfrequency, irregularly spaced, financial data.
A Continuous Time GARCH Process Driven by a Lévy Process: Stationarity and Second Order Behaviour
"... We use a discrete time analysis, giving necessary and sufficient conditions for the almost sure convergence of ARCH(1) and GARCH(1,1) discrete time models, to suggest an extension of the (G)ARCH concept to continuous time processes. Our “COGARCH” (continuous time GARCH) model, based on a single back ..."
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We use a discrete time analysis, giving necessary and sufficient conditions for the almost sure convergence of ARCH(1) and GARCH(1,1) discrete time models, to suggest an extension of the (G)ARCH concept to continuous time processes. Our “COGARCH” (continuous time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous time stochastic volatility models that have been proposed. The model generalises the essential features of discrete time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.
Managing Energy Price Risk (2 nd Edition) RISK Publications pp291304 (1999) Correlation and Cointegration in Energy Markets
"... Successful risk management requires real understanding of the nature of volatility and correlations between financial markets, and the problems inherent in calculating statistical estimates of these quantities. Whilst volatilities are based on the variances of individual returns distributions, corre ..."
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Successful risk management requires real understanding of the nature of volatility and correlations between financial markets, and the problems inherent in calculating statistical estimates of these quantities. Whilst volatilities are based on the variances of individual returns distributions, correlations depend on characteristics of the joint distributions between two related markets. This extra dimension adds a great deal of uncertainty to correlation risk measures. In fact, whilst it seems reasonable to assume that individual return processes are stationary, so that volatilities do exist, it is by no means always the case that two returns processes will be jointly stationary. So unconditional correlations may not even exist. Of course it is always possible to calculate a number that supposedly represents correlation, but often these numbers change considerably from day to day, a sign that the two returns processes are not jointly stationary. It is unfortunate that some standard correlation estimation methods induce an apparent stability that is purely an artefact of the method, and the true nature of underlying correlations is obscured. The first objective of this chapter is to review the different approaches to measuring
On the Covariance Matrices used in ValueatRisk Models
"... This paper examines the covariance matrices that are often used for internal valueatrisk models. We first show how the large covariance matrices necessary for global risk management systems can be generated using orthogonalization procedures in conjunction with univariate volatility forecasting me ..."
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This paper examines the covariance matrices that are often used for internal valueatrisk models. We first show how the large covariance matrices necessary for global risk management systems can be generated using orthogonalization procedures in conjunction with univariate volatility forecasting methods. We then examine the performance of three common volatility forecasting methods: the equally weighted average; the exponentially weighted average; and Generalised Autoregressive Conditional Heteroscedasticity (GARCH). Standard statistical evaluation criteria using equity and foreign exchange data with 1996 as the test period give mixed results, although generally favour the exponentially weighted moving average methodology for all but very short term holding periods. But these criteria assess the ability to model the centre of returns distributions, whereas valueatrisk models require accuracy in the tails. Operational evaluation takes the form of back testing volatility forecasts following the Bank for International Settlements (BIS) guidelines. For almost all major equity markets and US dollar exchange rates, both the equally weighted average and the GARCH models would be placed within the ‘green zone’. However on most of the test data, and particularly for foreign exchange, exponentially weighted moving average models predict an unacceptably high number of outliers. Thus valueatrisk measures calculated using this method would be understated.
Risk Management and Analysis: Measuring and Modelling Financial Risk (C. Alexander, Ed) Wileys, (1998) CHAPTER 4 Volatility and Correlation: Measurement, Models and Applications
"... The most widely accepted approach to ‘risk ’ in financial markets focuses on the measurement of volatility in certain returns distributions. 1 The volatility of portfolio returns depends on the variances and covariances between the risk factors of the portfolio, and the sensitivities of individual a ..."
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The most widely accepted approach to ‘risk ’ in financial markets focuses on the measurement of volatility in certain returns distributions. 1 The volatility of portfolio returns depends on the variances and covariances between the risk factors of the portfolio, and the sensitivities of individual assets to these risk factors. In linear portfolios, sensitivities are also measured by
Stationarity and Second Order Behaviour of Discrete and Continuous Time GARCH(1,1) Processes
"... We use a discrete time analysis, giving necessary and sufficient conditions for the almost sure convergence of ARCH(1) and GARCH(1,1) discrete time models, to suggest an extension of the (G)ARCH concept to continuous time processes. The models, based ..."
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We use a discrete time analysis, giving necessary and sufficient conditions for the almost sure convergence of ARCH(1) and GARCH(1,1) discrete time models, to suggest an extension of the (G)ARCH concept to continuous time processes. The models, based
Parameter Estimation of ARMA Models with GARCH/APARCH Errors
"... We report on concepts and methods to implement the family of ARMA models with GARCH/APARCH errors introduced by Ding, Granger and Engle. The software implementation is written in S and optimization of the constrained loglikelihood function is achieved with the help of a SQP solver. The implementat ..."
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We report on concepts and methods to implement the family of ARMA models with GARCH/APARCH errors introduced by Ding, Granger and Engle. The software implementation is written in S and optimization of the constrained loglikelihood function is achieved with the help of a SQP solver. The implementation is tested with Bollerslev’s GARCH(1,1) model applied to the DEMGBP foreign exchange rate data set given by Bollerslev and Ghysels. The results are compared with the benchmark implementation of Fiorentini, Calzolari and Panattoni. In addition the MA(1)APARCH(1,1) model for the SP500 stock market index analyzed by Ding, Granger and Engle is reestimated and compared with results obtained from the Ox/G@RCH and SPlus/Finmetrics software packages. The software is part of the Rmetrics open source project for computational finance and financial engineering. Implementations are available for both software environments, R and SPlus.