35464167
Spectral analysis of quadratic variation in the presence of market microstructure noise
We analyze the ex-post variation of equity prices in the frequency domain. A realized periodogram-based estimator is proposed, which consistently estimates the quadratic variation of the log equilibrium price process. For prices which are contaminated by market microstructure noise, the proposed estimator behaves like a filter: it removes the noise by filtering out high frequency periodograms. In other words, the proposed estimator converts high frequency data into low frequency periodograms. We show, through a simulation study and an application to the General Electric transaction prices, that the proposed estimator is insensitive to the choice of sampling frequency and it is competitive with other existing volatility measures.
17588636
ANSWERING THE SKEPTICS: YES, STANDARD VOLATILITY MODELS DO PROVIDE ACCURATE FORECASTS*
A voluminous literature has emerged for modeling the temporal dependencies in financial market volatility using ARCH and stochastic volatility models. While most of these studies have documented highly significant in-sample parameter estimates and pronounced intertemporal volatility persistence, traditional ex post forecast evaluation criteria suggest that the models provide seemingly poor volatility forecasts. Contrary to this contention, the authors show that volatility models produce strikingly accurate interdaily forecasts for the latent volatility factor that would be of interest in most financial applications. New methods for improved ex post interdaily volatility measurements based on high-frequency intradaily data are also discussed. Copyright 1998 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.
14366860
The Distribution of Exchange Rate Volatility
Using high-frequency data on Deutschemark and Yen returns against the dollar, we construct model-free estimates of daily exchange rate volatility and correlation, covering an entire decade. In addition to being model-free, our estimates are also approximately free of measurement error under general conditions, which we delineate. Hence, for all practical purposes, we can 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 normality-inducing volatility transformation, high contemporaneous correlation across volatilities, high correlation between correlation and volatilities, pronounced and highly persistent temporal variation in both volatilities and correlation, clear evidence of long-memory dynamics in both volatilities and correlation, and remarkably precise scaling laws under temporal aggregation.
14117713
A theoretical comparison between integrated and realized volatility
In this paper we provide both qualitative and quantitative measures of the precision of measuring integrated volatility by realized volatility for a fixed frequency of observation. We start by characterizing for a general diffusion the difference between realized and integrated volatility for a given frequency of observation. Then we compute the mean and variance of this noise and the correlation between the noise and the integrated volatility in the Eigenfunction Stochastic Volatility model of Meddahi (2001a). This model has as special cases log-normal, affine and GARCH diffusion models. Using previous empirical results, we show that the noise is substantial compared with the unconditional mean and variance of integrated volatility, even if one employs five-minute returns. We also propose a simple approach to capture the information about integrated volatility contained in the returns through the leverage effect. We show that in practice, the leverage effect does not matter. Copyright © 2002 John Wiley & Sons, Ltd.
17009113
Econometric analysis of realised volatility and its use in estimating stochastic volatility models
The availability of intra-data on the prices of speculative assets means that we can use quadratic variation like measures of activity in financial markets, called realised volatility, to study the stochastic properties of returns. Here we derive the moments and the asymptotic distribution of the realised volatility error - the difference between realised volatility and the actual volatility. These properties can be used to allow us to estimate the parameters of stochastic volatility models.
361689
Estimators of diffusions with randomly spaced discrete observations: A general theory
We provide a general method to analyze the asymptotic properties of a variety of estimators of continuous time diffusion processes when the data are not only discretely sampled in time but the time separating successive observations may possibly be random. We introduce a new operator, the generalized infinitesimal generator, to obtain Taylor expansions of the asymptotic moments of the estimators. As a special case, our results apply to the situation where the data are discretely sampled at a fixed nonrandom time interval. We include as specific examples estimators based on maximum-likelihood and discrete approximations such as the Euler scheme.
10474693
LIMIT THEOREMS FOR BIPOWER VARIATION IN FINANCIAL ECONOMETRICS
In this paper we provide an asymptotic analysis of generalized bipower measures of the variation of price processes in financial economics. These measures encompass the usual quadratic variation, power variation, and bipower variations that have been highlighted in recent years in financial econometrics. The analysis is carried out under some rather general Brownian semimartingale assumptions, which allow for standard leverage effects.Ole E. Barndorff-Nielsen's work is supported by the Centre for Analytical Finance (CAF), which is funded by the Danish Social Science Research Council. Neil Shephard's research is supported by the UK's ESRC through the grant “High frequency financial econometrics based upon power variation.” We thank the editor, Peter Phillips, and the referees for their stimulating comments on an earlier version.
54708131
Integrated volatility measuring from unevenly sampled observations
This paper derives the linear interpolation bias of realized volatility. To avoid the bias, the Fourier series estimator has been proposed by Malliavin and Mancino (2002). We examine the theoretical relationship between the Fourier estimator and realized volatility and show that the latter is the most efficient estimator in the class of the former.
8679740
ANOVA for diffusions and Itô processes
Ito processes are the most common form of continuous semimartingales, and include diffusion processes. This paper is concerned with the nonparametric regression relationship between two such Ito processes. We are interested in the quadratic variation (integrated volatility) of the residual in this regression, over a unit of time (such as a day). A main conceptual finding is that this quadratic variation can be estimated almost as if the residual process were observed, the difference being that there is also a bias which is of the same asymptotic order as the mixed normal error term. The proposed methodology, "ANOVA for diffusions and Ito processes," can be used to measure the statistical quality of a parametric model and, non-parametrically, the appropriateness of a one-regressor model in general. On the other hand, it also helps quantify and characterize the trading (hedging) error in the case of financial applications.
15908443
How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise
Classical statistics suggest that for inference purposes one should always use as much data as is available. We study how the presence of market microstructure noise in high-frequency financial data can change that result. We show that the optimal sampling frequency at which to estimate the parameters of a discretely sampled continuous-time model can be finite when the observations are contaminated by market microstructure effects. We then address the question of what to do about the presence of the noise. We show that modelling the noise term explicitly restores the first order statistical effect that sampling as often as possible is optimal. But, more surprisingly, we also demonstrate that this is true even if one misspecifies the assumed distribution of the noise term. Not only is it still optimal to sample as often as possible, but the estimator has the same variance as if the noise distribution had been correctly specified, implying that attempts to incorporate the noise into the analysis cannot do more harm than good. Finally, we study the same questions when the observations are sampled at random time intervals, which are an essential feature of transaction-level data.
40938303
Realized Variance and Market Microstructure Noise
We study market microstructure noise in high-frequency data and analyze its implications for the realized variance (RV) under a general specification for the noise. We show that kernel-based estimators can unearth important characteristics of market microstructure noise and that a simple kernel-based estimator dominates the RV for the estimation of integrated variance (IV). An empirical analysis of the Dow Jones Industrial Average stocks reveals that market microstructure noise is time-dependent and correlated with increments in the efficient price. This has important implications for volatility estimation based on high-frequency data. Finally, we apply cointegration techniques to decompose transaction prices and bid–ask quotes into an estimate of the efficient price and noise. This framework enables us to study the dynamic effects on transaction prices and quotes caused by changes in the efficient price.
2474282
Separating Microstructure Noise from Volatility
There are two volatility components embedded in the returns constructed using recorded stock prices: the genuine time-varying volatility of the unobservable returns that would prevail (in equilibrium) in a frictionless, full-information, economy and the variance of the equally unobservable microstructure noise. Using straightforward sample averages of high-frequency return data recorded at different frequencies, we provide a simple technique to identify both volatility features. We apply our methodology to a sample of S&P100 stocks.
73583893
Designing Realised Kernels to Measure the Ex-Post Variation of Equity Prices in the Presence of Noise
This paper shows how to use realised kernels to carry out efficient feasible inference on the expost variation of underlying equity prices in the presence of simple models of market frictions. The issue is subtle with only estimators which have symmetric weights delivering consistent estimators with mixed Gaussian limit theorems. The weights can be chosen to achieve the best possible rate of convergence and to have an asymptotic variance which is close to that of the maximum likelihood estimator in the parametric version of this problem. Realised kernels can also be selected to (i) be analysed using endogenously spaced data such as that in databases on transactions, (ii) allow for market frictions which are endogenous, (iii) allow for temporally dependent noise. The finite sample performance of our estimators is studied using simulation, while empirical work illustrates their use in practice.
154021931
High Frequency Data and Volatility in Foreign Exchange Rates
Exchange rates, like many other financial time series, display substantial heteroscedasticity. This poses obstacles in detecting trends and changes. Understanding volatility becomes extremely important in studying financial time series. Unfortunately, estimating volatility from low-frequency data, such as daily, weekly, or monthly observations, is very difficult. The recent availability of ultra-high-frequency observations, such as tick-by-tick data, to large financial institutions creates a new possibility for the analysis of volatile time series. This article uses tick-by-tick foreign-exchange rates to explore this new type of data. Unlike low-frequency data, high-frequency data have extremely high negative first-order autocorrelation in their return. In this article, I propose a model that can explain the negative autocorrelation and a volatility estimator for high-frequency data. The daily and hourly volatility estimates of exchange rate show some interesting patterns.
15862653
Realized volatility forecasting and market microstructure noise
We extend the analytical results for reduced form realized volatility based forecasting in ABM (2004) to allow for market microstructure frictions in the observed high-frequency returns. Our results build on the eigenfunction representation of the general stochastic volatility class of models developed byMeddahi (2001). In addition to traditional realized volatility measures and the role of the underlying sampling frequencies, we also explore the forecasting performance of several alternative volatility measures designed to mitigate the impact of the microstructure noise. Our analysis is facilitated by a simple unified quadratic form representation for all these estimators. Our results suggest that the detrimental impact of the noise on forecast accuracy can be substantial. Moreover, the linear forecasts based on a simple-to-implement 'average' (or 'subsampled') estimator obtained by averaging standard sparsely sampled realized volatility measures generally perform on par with the best alternative robust measures.
208056665
Multivariate Realised Kernels: Consistent Positive Semi-Definite Estimators of the Covariation of Equity Prices with Noise and Non-Synchronous Trading
We propose a multivariate realised kernel to estimate the ex -post covariation of log-prices. We show this new consistent estimator is guaranteed to be posit ive emi-definite and is robust to measurement noise of certain types and can also handle non-sync hronous trading. It is the first estimator which has these three properties which are all essential for empirical work in this area. We derive the large sample asymptotics of this estimator and assess it accuracy using a Monte Carlo study. We implement the estimator on some US equity data, comparing ou r res lts to previous work which has used returns measured over 5 or 10 minutes intervals. We show the new estimator is substantially more precise.
122810709
Spectral Analysis and Time Series
Preface. Preface to Volume 2. Contents of Volume 2. List of Main Notation. Basic Concepts. Elements of Probability Theory. Stationary Random Processes. Spectral Analysis. Estimation in the Time Domain. Estimation in the Frequency Domain. Spectral Analysis in Practice. Analysis of Processes with Mixed Spectra.
122793370
Asymptotic normality of spectral estimates
The asymptotic normality of some spectral estimates, including a functional central limit theorem for an estimate of the spectral distribution function, is proved for fourth-order stationary processes. In contrast to known results it is not assumed that all moments exist or that the process is linear. The data are allowed to be tapered. Using some recent results on the central limit theorem for stationary processes, corollaries are obtained for strong and [phi]-mixing sequences and linear transformations of martingale differences.
126083593
Microstructure Noise in the Continuous Case: The Pre-Averaging Approach ∗
This paper presents a generalized pre-averaging approach for estimating the integrated volatility. This approach also provides consistent estimators of other powers of volatility – in particular, it gives feasible ways to consistently estimate the asymptotic variance of the estimator of the integrated volatility. We show that our approach, which possess an intuitive transparency, can generate rate optimal estimators (with convergence rate n 1/4 ).
38027967
Fourier series method for measurement of multivariate volatilities
Abstract. We present a methodology based on Fourier series analysis to compute time series volatility when the data are observations of a semimartingale. The procedure is not based on the Wiener theorem for the quadratic variation, but on the computation of the Fourier coefficients of the process and therefore it relies on the integration of the time series rather than on its differentiation. The method is fully model free and nonparametric. These features make the method well suited for financial market applications, and in particular for the analysis of high frequency time series and for the computation of cross volatilities.
121208188
Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics
Non-Gaussian processes of Ornstein–Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory.
16147780
Alternative models for stock price dynamics
This paper evaluates the role of various volatility specifications, such as multiple stochastic volatility (SV) factors and jump components, in appropriate modeling of equity return distributions. We use estimation technology that facilitates non-nested model comparisons and use a long data set which provides rich information about the conditional and unconditional distribution of returns. We consider two broad families of models: (1) the multifactor loglinear family, and (2) the affine-jump family. Both classes of models have attracted much attention in the derivatives and econometrics literatures. There are various trade-offs in considering such diverse specifications. If pure diffusion SV models are chosen over jump diffusions, it has important implications for hedging strategies. If logaritmic models are chosen over affine ones, it may seriously complicate option pricing. Comparing many different specifications of pure diffusion multi-factor models and jump diffusion models, we find that (1) log linear models have to be extented to 2 factors with feedback in the mean reverting factor, (2) affine models have to have a jumps in returns, stochastic volatility and probably both. Models (1) and (2) are observationally equivalent on the data set in hand. In either (1) or (2) the key is that the volatility can move violently. As we obtain models with comparable empirical fit, one must make a choice based on arguments other than statistical goodness of fit criteria. The considerations include facility to price options, to hedge and parsimony. The affine specification with jumps in volatility might therefore be preferred because of the closed-form derivatives prices. Nous examinons un ensemble de diffusions avec volatilite stochastique et de sauts afin de modeliser la distribution des rendements d'actifs boursiers. Puisque certains modeles sont non-emboites, nous utilisons la methode EMM afin d'etudier et de comparer le comportement des differents modeles.
9089417
The Realized Laplace Transform of Volatility
We introduce a new measure constructed from high-frequency financial data which we call the Realized Laplace Transform of volatility. The statistic provides a nonparametric estimate for the empirical Laplace transform of the latent stochastic volatility process over a given interval of time. When a long span of data is used, i.e., under joint long-span and fill-in asymptotics, it is an estimate of the volatility Laplace transform. The asymptotic behavior of the statistic depends on the small scale behavior of the driving martingale. We derive the asymptotics both in the case when the latter is known and when it needs to be inferred from the data. When the underlying process is a jump-diffusion our statistic is robust to jumps and when the process is pure-jump it is robust to presence of less active jumps. We apply our results to simulated and real financial data.
154050020
Forecasting financial market volatility: Sample frequency vis-a-vis forecast horizon
Abstract This paper explores the return volatility predictability inherent in high-frequency speculative returns. Our analysis focuses on a refinement of the more traditional volatility measures, the integrated volatility , which links the notion of volatility more directly to the return variance over the relevant horizon. In our empirical analysis of the foreign exchange market the integrated volatility is conveniently approximated by a cumulative sum of the squared intraday returns. Forecast horizons ranging from short intraday to 1-month intervals are investigated. We document that standard volatility models generally provide good forecasts of this economically relevant volatility measure. Moreover, the use of high-frequency returns significantly improves the longer run interdaily volatility forecasts, both in theory and practice. The results are thus directly relevant for general research methodology as well as industry applications.
17992188
The Relative Contribution of Jumps to Total Price Variance
We examine tests for jumps based on recent asymptotic results; we interpret the tests as Hausman-type tests. Monte Carlo evidence suggests that the daily ratio z-statistic has appropriate size, good power, and good jump detection capabilities revealed by the confusion matrix comprised of jump classification probabilities. We identify a pitfall in applying the asymptotic approximation over an entire sample. Theoretical and Monte Carlo analysis indicates that microstructure noise biases the tests against detecting jumps, and that a simple lagging strategy corrects the bias. Empirical work documents evidence for jumps that account for 7% of stock market price variance. Copyright 2005, Oxford University Press.
9674727
Power and bipower variation with stochastic volatility and jumps
This paper shows that realised power variation and its extension we introduce here called realised bipower variation is somewhat robust to rare jumps. We show realised bipower variation estimates integrated variance in SV models --- thus providing a model free and consistent alternative to realised variance. Its robustness property means that if we have an SV plus infrequent jumps process then the difference between realised variance and realised bipower variation estimates the quadratic variation of the jump component. This seems to be the first method which can divide up quadratic variation into its continuous and jump components. Various extensions are given. Proofs of special cases of these results are given. Detailed mathematical results will be reported elsewhere.
120991209
Realized Kernels in Practice: Trades and Quotes
Realized kernels use high-frequency data to estimate daily volatility of individual stock prices. They can be applied to either trade or quote data. Here we provide the details of how we suggest implementing them in practice. We compare the estimates based on trade and quote data for the same stock and find a remarkable level of agreement. Copyright The Author(s). Journal compilation Royal Economic Society 2009