On the realized volatility of the ECX CO2 emissions 2008 futures contract: distribution, dynamics and forecasting

The recent implementation of the EU Emissions Trading Scheme (EU ETS) in January 2005 created new financial risks for emitting firms. To deal with these risks, options are traded since October 2006. Because the EU ETS is a new market, the relevant underlying model for option pricing is still a controversial issue. This article improves our understanding of this issue by characterizing the conditional and unconditional distributions of the realized volatility for the 2008 futures contract in the European Climate Exchange (ECX), which is valid during Phase II (2008-2012) of the EU ETS. The realized volatility measures from naive, kernel-based and subsampling estimators are used to obtain inferences about the distributional and dynamic properties of the ECX emissions futures volatility. The distribution of the daily realized volatility in logarithmic form is shown to be close to normal. The mixture-of-distributions hypothesis is strongly rejected, as the returns standardized using daily measures of volatility clearly departs from normality. A simplified HAR-RV model (Corsi, 2009) with only a weekly component, which reproduces long memory properties of the series, is then used to model the volatility dynamics. Finally, the predictive accuracy of the HAR-RV model is tested against GARCH specifications using one-step-ahead forecasts, which confirms the HAR-RV superior ability. Our conclusions indicate that (i) the standard Brownian motion is not an adequate tool for option pricing in the EU ETS, and (ii) a jump component should be included in the stochastic process to price options, thus providing more efficient tools for risk-management activities.CO2 Price; Realized Volatility; HAR-RV; GARCH; Futures Trading; Emissions Markets; EU ETS; Intraday data; Forecasting

On the realized volatility of the ECX CO2 emissions 2008 futures contract: distribution, dynamics and forecasting

The recent implementation of the EU Emissions Trading Scheme (EU ETS) in January 2005 created new financial risks for emitting firms. To deal with these risks, options are traded since October 2006. Because the EU ETS is a new market, the relevant underlying model for option pricing is still a controversial issue. This article improves our understanding of this issue by characterizing the conditional and unconditional distributions of the realized volatility for the 2008 futures contract in the European Climate Exchange (ECX), which is valid during Phase II (2008-2012) of the EU ETS. The realized volatility measures from naive, kernel-based and subsampling estimators are used to obtain inferences about the distributional and dynamic properties of the ECX emissions futures volatility. The distribution of the daily realized volatility in logarithmic form is shown to be close to normal. The mixture-of-distributions hypothesis is strongly rejected, as the returns standardized using daily measures of volatility clearly departs from normality. A simplified HAR-RV model (Corsi, 2009) with only a weekly component, which reproduces long memory properties of the series, is then used to model the volatility dynamics. Finally, the predictive accuracy of the HAR-RV model is tested against GARCH specifications using one-step-ahead forecasts, which confirms the HAR-RV superior ability. Our conclusions indicate that (i) the standard Brownian motion is not an adequate tool for option pricing in the EU ETS, and (ii) a jump component should be included in the stochastic process to price options, thus providing more efficient tools for risk-management activities.CO2 Price, Realized Volatility, HAR-RV, GARCH, Futures Trading, Emissions Markets, EU ETS, Intraday data, Forecasting

The volatility of realized volatility

Using unobservable conditional variance as measure, latent-variable approaches, such as GARCH and stochastic-volatility models, have traditionally been dominating the empirical finance literature. In recent years, with the availability of high-frequency financial market data modeling realized volatility has become a new and innovative research direction. By constructing "observable" or realized volatility series from intraday transaction data, the use of standard time series models, such as ARFIMA models, have become a promising strategy for modeling and predicting (daily) volatility. In this paper, we show that the residuals of the commonly used time-series models for realized volatility exhibit non-Gaussianity and volatility clustering. We propose extensions to explicitly account for these properties and assess their relevance when modeling and forecasting realized volatility. In an empirical application for S&P500 index futures we show that allowing for time-varying volatility of realized volatility leads to a substantial improvement of the model's fit as well as predictive performance. Furthermore, the distributional assumption for residuals plays a crucial role in density forecasting. Klassifikation: C22, C51, C52, C5

Combining long memory and level shifts in modeling and forecasting the volatility of asset returns

We propose a parametric state space model of asset return volatility with an accompanying estimation and forecasting framework that allows for ARFIMA dynamics, random level shifts and measurement errors. The Kalman filter is used to construct the state-augmented likelihood function and subsequently to generate forecasts, which are mean- and path-corrected. We apply our model to eight daily volatility series constructed from both high-frequency and daily returns. Full sample parameter estimates reveal that random level shifts are present in all series. Genuine long memory is present in high-frequency measures of volatility whereas there is little remaining dynamics in the volatility measures constructed using daily returns. From extensive forecast evaluations, we find that our ARFIMA model with random level shifts consistently belongs to the 10% Model Confidence Set across a variety of forecast horizons, asset classes, and volatility measures. The gains in forecast accuracy can be very pronounced, especially at longer horizons

Combining long memory and level shifts in modeling and forecasting the volatility of asset returns

We propose a parametric state space model of asset return volatility with an accompanying estimation and forecasting framework that allows for ARFIMA dynamics, random level shifts and measurement errors. The Kalman filter is used to construct the state-augmented likelihood function and subsequently to generate forecasts, which are mean and path-corrected. We apply our model to eight daily volatility series constructed from both high-frequency and daily returns. Full sample parameter estimates reveal that random level shifts are present in all series. Genuine long memory is present in most high-frequency measures of volatility, whereas there is little remaining dynamics in the volatility measures constructed using daily returns. From extensive forecast evaluations, we find that our ARFIMA model with random level shifts consistently belongs to the 10% Model Confidence Set across a variety of forecast horizons, asset classes and volatility measures. The gains in forecast accuracy can be very pronounced, especially at longer horizons

A multiple regime smooth transition heterogeneous autoregressive model for long memory and asymmetries

In this paper we propose a flexible model to capture nonlinearities and long-range dependence in time series dynamics. The new model is a multiple regime smooth transition extension of the Heterogenous Autoregressive (HAR) model, which is specifically designed to model the behavior of the volatility inherent in financial time series. The model is able to describe simultaneously long memory, as well as sign and size asymmetries. A sequence of tests is developed to determine the number of regimes, and an estimation and testing procedure is presented. Monte Carlo simulations evaluate the finite-sample properties of the proposed tests and estimation procedures. We apply the model to several Dow Jones Industrial Average index stocks using transaction level data from the Trades and Quotes database that covers ten years of data. We find strong support for long memory and both sign and size asymmetries. Furthermore, the new model, when combined with the linear HAR model, is viable and flexible for purposes of forecasting volatility.Realized volatility, smooth transition, heterogeneous autoregression, financial econometrics,leverage, sign and size asymmetries, forecasting, risk management, model combination.

The Volatility of Realized Volatility

Using unobservable conditional variance as measure, latent–variable approaches, such as GARCH and stochastic–volatility models, have traditionally been dominating the empirical finance literature. In recent years, with the availability of high–frequency financial market data modeling realized volatility has become a new and innovative research direction. By constructing “observable” or realized volatility series from intraday transaction data, the use of standard time series models, such as ARFIMA models, have become a promising strategy for modeling and predicting (daily) volatility. In this paper, we show that the residuals of the commonly used time–series models for realized volatility exhibit non–Gaussianity and volatility clustering. We propose extensions to explicitly account for these properties and assess their relevance when modeling and forecasting realized volatility. In an empirical application for S&P500 index futures we show that allowing for time–varying volatility of realized volatility leads to a substantial improvement of the model’s fit as well as predictive performance. Furthermore, the distributional assumption for residuals plays a crucial role in density forecasting.Finance, Realized Volatility, Realized Quarticity, GARCH, Normal Inverse Gaussian Distribution, Density Forecasting

Roughing it Up: Including Jump Components in the Measurement, Modeling and Forecasting of Return Volatility

A rapidly growing literature has documented important improvements in financial return volatility measurement and forecasting via use of realized variation measures constructed from high-frequency returns coupled with simple modeling procedures. Building on recent theoretical results in Barndorff-Nielsen and Shephard (2004a, 2005) for related bi-power variation measures, the present paper provides a practical and robust framework for non-parametrically measuring the jump component in asset return volatility. In an application to the DM/$ exchange rate, the S&P500 market index, and the 30-year U.S. Treasury bond yield, we find that jumps are both highly prevalent and distinctly less persistent than the continuous sample path variation process. Moreover, many jumps appear directly associated with specific macroeconomic news announcements. Separating jump from non-jump movements in a simple but sophisticated volatility forecasting model, we find that almost all of the predictability in daily, weekly, and monthly return volatilities comes from the non-jump component. Our results thus set the stage for a number of interesting future econometric developments and important financial applications by separately modeling, forecasting, and pricing the continuous and jump components of the total return variation process.

Some Like it Smooth, and Some Like it Rough: Untangling Continuous and Jump Components in Measuring, Modeling, and Forecasting Asset Return Volatility

A rapidly growing literature has documented important improvements in volatility measurement and forecasting performance through the use of realized volatilities constructed from high frequency returns coupled with relatively simple reduced-form time series modeling procedures. Building on recent theoretical results from Barndorff-Nielsen and Shephard (2003c,d) for related bi-power variation measures involving the sum of high-frequency absolute returns, the present paper provides a practical framework for non-parametrically measuring the jump component in realized volatility measurements. Exploiting these ideas for a decade of high-frequency five-minute returns for the DM/$ exchange rate, the S&P500 market index, and the 30-year U.S. Treasury bond yield, we find the jump component of the price process to be distinctly less persistent than the continuous sample path component. Explicitly including the jump measure as an additional explanatory variable in an easy-to implement reduced form model for realized volatility results in highly significant jump coefficient estimates at the daily, weekly and quarterly forecast horizons.Continuous-time methods, jumps, quadratic variation, realized volatility, bi-power variation, high-frequency data, volatility forecasting, HAR-RV model

Modeling and predicting the CBOE market volatility index

This paper performs a thorough statistical examination of the time-series properties of the market volatility index (VIX) from the Chicago Board Options Exchange (CBOE). The motivation lies on the widespread consensus that the VIX is a barometer to the overall market sentiment as to what concerns risk appetite. To assess the statistical behavior of the time series, we run a series of preliminary analyses whose results suggest there is some long-range dependence in the VIX index. This is consistent with the strong empirical evidence in the literature supporting long memory in both options-implied and realized volatilities. We thus resort to linear and nonlinear heterogeneous autoregressive (HAR) processes, including smooth transition and threshold HAR-type models, as well as to smooth transition autoregressive trees (START) for modeling and forecasting purposes. The in-sample results for the HAR-type indicate that they cope with the long-range dependence in the VIX time series as well as the more popular ARFIMA model. In addition, the highly nonlinear START specification also does a god job in controlling for the long memory. The out-of-sample analysis evince that the linear ARMA and ARFIMA models perform very well in the short run and very poorly in the long-run, whereas the START model entails by far the best results for the longer horizon despite of failing at shorter horizons. In contrast, the HAR-type models entail reasonable relative performances in most horizons. Finally, we also show how a simple forecast combination brings about great improvements in terms of predictive ability for most horizons.heterogeneous autoregression, implied volatility, smooth transition, VIX.