
on Financial Markets 
By:  Thomas Busch; Thomas Busch; Bent Jesper Christensen; Morten Ørregaard Nielsen (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  We study the forecasting of future realized volatility in the stock, bond, and for eign exchange markets, as well as the continuous sample path and jump components of this, from variables in the information set, including implied volatility backed out from option prices. Recent nonparametric statistical techniques of Barndor¤Nielsen & Shephard (2004, 2006) are used to separate realized volatility into its continuous and jump components, which enhances forecasting performance, as shown by Andersen, Bollerslev & Diebold (2005). We generalize the heterogeneous autoregressive (HAR) model of Corsi (2004) to include implied volatility as an additional regressor, and to the separate forecasting of the realized components. We also introduce a new vector HAR (VecHAR) model for the resulting simultaneous system, controlling for possible endogeneity issues in the forecasting equations. We show that implied volatility con tains incremental information about future volatility relative to both continuous and jump components of past realized volatility. Indeed, in the foreign exchange market, implied volatility completely subsumes the information content of daily, weekly, and monthly realized volatility measures, when forecasting future realized volatility or its continuous component. In addition, implied volatility is an unbiased forecast of future realized volatility in the foreign exchange and stock markets. Perhaps surprisingly, the jump component of realized return volatility is, to some extent, predictable, and options appear to be calibrated to incorporate information about future jumps in all three markets. 
Keywords:  Bipower variation, HAR, Heterogeneous Autoregressive Model, implied volatility, jumps, options, realized volatility, VecHAR, volatility forecasting 
JEL:  C22 C32 F31 G1 
Date:  2007–06–06 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200709&r=fmk 
By:  Bent Jesper Christensen; Morten Ørregaard Nielsen; Jie Zhu (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  We extend the fractionally integrated exponential GARCH (FIEGARCH) model for daily stock return data with long memory in return volatility of Bollerslev and Mikkelsen (1996) by introducing a possible volatilityinmean effect. To avoid that the long memory property of volatility carries over to returns, we consider a filtered FIEGARCHinmean (FIEGARCHM) effect in the return equation. The filtering of the volatilityinmean component thus allows the coexistence of long memory in volatility and short memory in returns. We present an application to the S&P 500 index which documents the empirical relevance of our model. 
Keywords:  FIEGARCH, financial leverage, GARCH, long memory, riskreturn tradeoff, stock returns, volatility feedback 
JEL:  C22 
Date:  2007–06–12 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200710&r=fmk 
By:  Tim Bollerslev; Michael Gibson; Hao Zhou (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  This paper proposes a method for constructing a volatility risk premium, or investor risk aversion, index. The method is intuitive and simple to implement, relying on the sample moments of the recently popularized modelfree realized and optionimplied volatility measures. A smallscale Monte Carlo experiment confirms that the procedure works well in practice. Implementing the procedure with actual S&P500 optionimplied volatilities and highfrequency fiveminutebased realized volatilities indicates significant temporal dependencies in the estimated stochastic volatility risk premium, which we in turn relate to a set of macrofinance state variables. We also find that the extracted volatility risk premium helps predict future stock market returns. 
Keywords:  Stochastic Volatility Risk Premium, ModelFree Implied Volatility, ModelFree Realized Volatility, BlackScholes, GMM Estimation, Return Predictability 
JEL:  G12 G13 C51 C52 
Date:  2007–08–16 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200716&r=fmk 
By:  Torben G. Andersen; Tim Bollerslev; Francis X. Diebold; Clara Vega (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  Using a unique highfrequency futures dataset, we characterize the response of U.S., German and British stock, bond and foreign exchange markets to realtime U.S. macroeconomic news. We find that news produces conditional mean jumps, hence highfrequency stock, bond and exchange rate dynamics are linked to fundamentals. Equity markets, moreover, react differently to news depending on the stage of the business cycle, which explains the low correlation between stock and bond returns when averaged over the cycle. Hence our results qualify earlier work suggesting that bond markets react most strongly to macroeconomic news, in particular, when conditioning on the state of the economy, the equity and foreign exchange markets appear equally responsive. Finally, we also document important contemporaneous links across all markets and countries, even after controlling for the effects of macroeconomic news. 
Keywords:  Asset Pricing, Macroeconomic News Announcements, Financial Market Linkages, Market Microstructure, HighFrequency Data, Survey Data, Asset Return Volatility, Forecasting 
JEL:  F3 F4 G1 C5 
Date:  2007–08–16 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200720&r=fmk 
By:  Torben G. Andersen; Tim Bollerslev; Per Houmann Frederiksen; Morten Ørregaard Nielsen (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  We provide an empirical framework for assessing the distributional properties of daily specu lative returns within the context of the continuoustime modeling paradigm traditionally used in asset pricing finance. Our approach builds directly on recently developed realized variation measures and nonparametric jump detection statistics constructed from highfrequency intra day data. A sequence of relatively simpletoimplement momentbased tests involving various transforms of the daily returns speak directly to the import of different features of the under lying continuoustime processes that might have generated the data. As such, the tests may serve as a useful diagnostic tool in the specification of empirically more realistic asset pricing models. Our results are also directly related to the popular mixtureofdistributions hypoth esis and the role of the corresponding latent information arrival process. On applying our sequential test procedure to the thirty individual stocks in the Dow Jones Industrial Average index, the data suggest that it is important to allow for both timevarying diffusive volatility, jumps, and leverage effects in order to satisfactorily describe the daily stock price dynamics. At a broader level, the empirical results also illustrate how the realized variation measures and highfrequency sampling schemes may be used in eliciting important distributional features and asset pricing implications more generally. 
Keywords:  Return distributions, continuoustime models, mixtureofdistributions hypothesis, financialtime sampling, highfrequency data, volatility signature plots, realized volatilities, jumps, leverage and volatility feedback effects 
JEL:  C1 G1 
Date:  2007–08–16 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200721&r=fmk 
By:  Tim Bollerslev; Uta Kretschmer; Christian Pigorsch; George Tauchen (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  We develop an empirically highly accurate discretetime daily stochastic volatility model that explicitly distinguishes between the jump and continuoustime components of price movements using nonparametric realized variation and Bipower variation measures constructed from highfrequency intraday data. The model setup allows us to directly assess the structural interdependencies among the shocks to returns and the two different volatility components. The model estimates suggest that the leverage effect, or asymmetry between returns and volatility, works primarily through the continuous volatility component. The excellent fit of the model makes it an ideal candidate for an easyto implement auxiliary model in the context of indirect estimation of empirically more realistic continuoustime jump diffusion and L´evydriven stochastic volatility models, effectively incorporating the interdaily dependencies inherent in the highfrequency intraday data. 
Keywords:  Realized volatility, Bipower variation, Jumps, Leverage effect, Simultaneous equation model 
JEL:  C1 C3 C5 G1 
Date:  2007–08–16 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200722&r=fmk 
By:  Torben G. Andersen; Oleg Bondarenko (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  The notion of modelfree implied volatility (MFIV), constituting the basis for the highly publicized VIX volatility index, can be hard to measure with accuracy due to the lack of precise prices for options with strikes in the tails of the return distribution. This is reflected in practice as the VIX index is computed through a tailtruncation which renders it more compatible with the related concept of corridor implied volatility (CIV). We provide a comprehensive derivation of the CIV measure and relate it to MFIV under general assumptions. In addition, we price the various volatility contracts, and hence estimate the corresponding volatility measures, under the standard BlackScholes model. Finally, we undertake the first empirical exploration of the CIV measures in the literature. Our results indicate that the measure can help us refine and systematize the information embedded in the derivatives markets. As such, the CIV measure may serve as a tool to facilitate empirical analysis of both volatility forecasting and volatility risk pricing across distinct future states of the world for diverse asset categories and time horizons. 
Keywords:  ModelFree Implied Volatility, Corridor Implied Volatility, Realized Volatility, VIX, Volatility Forecasting, RiskNeutral Density 
JEL:  C53 G12 G13 
Date:  2007–09–17 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200724&r=fmk 
By:  Torben G. Andersen; Luca Benzoni (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  We investigate whether bonds span the volatility risk in the U.S. Treasury market, as predicted by most `a±ne' term structure models. To this end, we construct powerful and modelfree empirical measures of the quadratic yield variation for a crosssection of ¯xedmaturity zerocoupon bonds (`realized yield volatility') through the use of highfrequency data. We ¯nd that the yield curve fails to span yield volatility, as the systematic volatility factors are largely unrelated to the cross section of yields. We conclude that a broad class of a±ne di®usive, Gaussianquadratic and a±ne jumpdi®usive models is incapable of accommodating the observed yield volatility dynamics. An important implication is that the bond markets per se are incomplete and yield volatility risk cannot be hedged by taking positions solely in the Treasury bond market. We also advocate using the empirical realized yield volatility measures more broadly as a basis for speci¯cation testing and (parametric) model selection within the term structure literature. 
Date:  2007–09–17 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200725&r=fmk 
By:  Mark Podolskij; Daniel Ziggel (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  We propose a new test for the parametric form of the volatility function in continuous time diffusion models of the type dXt = a(t,Xt)dt + s(t,Xt)dWt. Our approach involves a rangebased estimation of the integrated volatility and the integrated quarticity, which are used to construct the test statistic. Under rather weak assumptions on the drift and volatility we prove weak convergence of the test statistic to a centered mixed Gaussian distribution. As a consequence we obtain a test, which is consistent for any fixed alternative. Moreover, we present a parametric bootstrap procedure which provides a better approximation of the distribution of the test statistic. Finally, it is demonstrated by means of Monte Carlo study that the rangebased test is more powerful than the returnbased test when comparing at the same sampling frequency. 
Keywords:  Bipower Variation, Central Limit Theorem, Diffusion Models, GoodnessOf Fit Testing, HighFrequency Data, Integrated Volatility, RangeBased Bipower Variation, Semimartingale Theory 
JEL:  C12 C14 
Date:  2007–09–19 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200726&r=fmk 
By:  Peter Christoffersen; Kris Jacobs; Karim Mimouni (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  Most recent empirical option valuation studies build on the affine square root (SQR) stochastic volatility model. The SQR model is a convenient choice, because it yields closedform solutions for option prices. However, relatively little is known about the resulting biases. We investigate alternatives to the SQR model, by comparing its empirical performance with that of five different but equally parsimonious stochastic volatility models. We provide empirical evidence from three different sources. We first use realized volatilities to assess the properties of the SQR model and to guide us in the search for alternative specifications. We then estimate the models using maximum likelihood on S&P500 returns. Finally, we employ nonlinear least squares on a panel of option data. In comparison with earlier studies that explicitly solve the filtering problem, we analyze a more comprehensive option data set. The scope of our analysis is feasible because of our use of the particle filter. The three sources of data we employ all point to the same conclusion: the SQR model is misspecified. Overall, the best of the alternative volatility specifications is a model with linear rather than square root diffusion for variance which we refer to as the VAR model. This model captures the stylized facts in realized volatilities, it performs well in fitting various samples of index returns, and it has the lowest option implied volatility mean squared errors in and outofsample. 
Keywords:  Stochastic volatility, option valuation, particle filtering, skewness, kurtosis, mean reversion 
JEL:  G12 
Date:  2007–11–15 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200737&r=fmk 
By:  Peter Christoffersen; Kris Jacobs; Gregory Vainberg (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  Few issues are more important for finance practice than the computation of market betas. Existing approaches compute market betas using historical data. While these approaches differ in terms of statistical sophistication and the modeling of the timevariation in the betas, they are all backwardlooking. This paper introduces a radically different approach to estimating market betas. Using the tools in Bakshi and Madan (2000) and Bakshi, Kapadia and Madan (2003) we employ the information embedded in the prices of individual stock options and index options to compute our forwardlooking market beta at the daily frequency. This beta can be computed using option data for a single day, and is able to reflect sudden changes in the structure of the underlying company. Based on an empirical investigation of daily crosssections of option contracts on thirty underlying companies, we conclude that these forwardlooking betas contain information relevant for forecasting future betas that is not contained in historical betas. 
Keywords:  market beta, CAPM, historical, forwardlooking, optionimplied, capital budgeting, event studies, modelfree moments 
JEL:  G12 
Date:  2007–11–22 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200739&r=fmk 
By:  Peter Christoffersen; Kris Jacobs; Chayawat Ornthanalai; Yintian Wang (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  This paper presents a new model for the valuation of European options, in which the volatility of returns consists of two components. One of these components is a longrun component, and it can be modeled as fully persistent. The other component is shortrun and has a zero mean. Our model can be viewed as an affine version of Engle and Lee (1999), allowing for easy valuation of European options. The model substantially outperforms a benchmark singlecomponent volatility model that is wellestablished in the literature, and it fits options better than a model that combines conditional heteroskedasticity and Poissonnormal jumps. The component model’s superior performance is partly due to its improved ability to model the smirk and the path of spot volatility, but its most distinctive feature is its ability to model the volatility term structure. This feature enables the component model to jointly model longmaturity and shortmaturity options. 
Keywords:  Volatility term structure; GARCH; outofsample 
JEL:  G12 
Date:  2008–02–18 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200811&r=fmk 
By:  Lars Stentoft (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  In the present paper we suggest to model Realized Volatility, an estimate of daily volatility based on high frequency data, as an Inverse Gaussian distributed variable with time varying mean, and we examine the joint properties of Realized Volatility and asset returns. We derive the appropriate dynamics to be used for option pricing purposes in this framework, and we show that our model explains some of the mispricings found when using traditional option pricing models based on interdaily data. We then show explicitly that a Generalized Autoregressive Conditional Heteroskedastic model with Normal Inverse Gaussian distributed innovations is the corresponding benchmark model when only daily data is used. Finally, we perform an empirical analysis using stock options for three large American companies, and we show that in all cases our model performs significantly better than the corresponding benchmark model estimated on return data alone. Hence the paper provides evidence on the value of using high frequency data for option pricing purposes. 
Keywords:  Option Pricing, Realized Volatility, Stochastic Volatility, GARCH 
JEL:  C22 C53 G13 
Date:  2008–03–03 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200813&r=fmk 
By:  Jie Zhu (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  In this paper a twocomponent volatility model based on the component's first moment is introduced to describe the dynamic of speculative return volatility. The two components capture the volatile and persistent part of volatility respectively. Then the model is applied to 10 AsiaPacific stock markets. Their inmean effects on return are also tested. The empirical results show that the persistent component accounts much more for volatility dynamic process than the volatile component. However the volatile component is found to be a significant pricing factor of asset returns for most markets, a positive or riskpremium effect exists between return and the volatile component, yet the persistent component is not significantly priced for return dynamic process. 
Keywords:  Risk, Return, Inmean effect, Volatile, Persistent, Innovations 
JEL:  C14 G12 G15 
Date:  2008–03–05 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200814&r=fmk 
By:  Viktor Todorov; Tim Bollerslev (School of Economics and Management, University of Aarhus, Denmark) 
Abstract:  We provide a new theoretical framework for disentangling and estimating sensitivity towards systematic diffusive and jump risks in the context of factor pricing models. Our estimates of the sensitivities towards systematic risks, or betas, are based on the notion of increasingly finer sampled returns over fixed time intervals. In addition to establish ing consistency of our estimators, we also derive Central Limit Theorems characterizing their asymptotic distributions. In an empirical application of the new procedures using highfrequency data for forty individual stocks and an aggregate market portfolio, we find the estimated diffusive and jump betas with respect to the market to be quite dif ferent for many of the stocks. Our findings have direct and important implications for empirical asset pricing finance and practical portfolio and risk management decisions. 
Keywords:  Factor models, systematic risk, common jumps, highfrequency data, realized variation 
JEL:  C13 C14 G10 G12 
Date:  2007–08–16 
URL:  http://d.repec.org/n?u=RePEc:aah:create:200715&r=fmk 