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24
Rangebased estimation of stochastic volatility models
, 2002
"... We propose using the price range in the estimation of stochastic volatility models. We show theoretically, numerically, and empirically that rangebased volatility proxies are not only highly efficient, but also approximately Gaussian and robust to microstructure noise. Hence rangebased Gaussian qu ..."
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Cited by 117 (11 self)
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We propose using the price range in the estimation of stochastic volatility models. We show theoretically, numerically, and empirically that rangebased volatility proxies are not only highly efficient, but also approximately Gaussian and robust to microstructure noise. Hence rangebased Gaussian quasimaximum likelihood estimation produces highly efficient estimates of stochastic volatility models and extractions of latent volatility. We use our method to examine the dynamics of daily exchange rate volatility and find the evidence points strongly toward twofactor models with one highly persistent factor and one quickly meanreverting factor. VOLATILITY IS A CENTRAL CONCEPT in finance, whether in asset pricing, portfolio choice, or risk management. Not long ago, theoretical models routinely assumed constant volatility ~e.g., Merton ~1969!, Black and Scholes ~1973!!. Today, however, we widely acknowledge that volatility is both time varying and predictable ~e.g., Andersen and Bollerslev ~1997!!, andstochastic volatility models are commonplace. Discrete and continuoustime stochastic volatility models are extensively used in theoretical finance, empirical finance, and financial econometrics, both in academe and industry ~e.g., Hull and
Efficient estimation of stochastic volatility using noisy observations: A multiscale approach
, 2004
"... With the availability of high frequency financial data, nonparametric estimation of volatility of an asset return process becomes feasible. A major problem is how to estimate the volatility consistently and efficiently, when the observed asset returns contain error or noise, for example, in the form ..."
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Cited by 85 (10 self)
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With the availability of high frequency financial data, nonparametric estimation of volatility of an asset return process becomes feasible. A major problem is how to estimate the volatility consistently and efficiently, when the observed asset returns contain error or noise, for example, in the form of microstructure noise. The former (consistency) has been addressed heavily in the recent literature, however, the resulting estimator is not quite efficient. In Zhang, Mykland, and AïtSahalia (2003), the best estimator converges to the true volatility only at the rate of n −1/6. In this paper, we propose an efficient estimator which converges to the true at the rate of n −1/4, which is the best attainable. The estimator remains valid when the observation noise is dependent. Some key words and phrases: consistency, dependent noise, discrete observation, efficiency, Ito process, microstructure noise, observation error, rate of convergence, realized volatility
A DiscreteTime Model for Daily S&P500 Returns and Realized Variations: Jumps and Leverage Effects
, 2007
"... 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 dat ..."
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Cited by 20 (1 self)
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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 easytoimplement auxiliary model in the context of indirect estimation of empirically more realistic continuoustime jump diffusion and Lévydriven stochastic volatility models, effectively incorporating the interdaily dependencies inherent in the highfrequency intraday data.
Variational Sums and Power Variation: a unifying approach to model selection and estimation in semimartingale models
 Statistics & Decisions
, 2003
"... In the framework of general semimartingale models we provide limit theorems for variational sums including the pth power variation, i.e. the sum of pth absolute powers of increments of a process. This gives new insight in the use of quadratic and realised power variation as an estimate for the ..."
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Cited by 15 (1 self)
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In the framework of general semimartingale models we provide limit theorems for variational sums including the pth power variation, i.e. the sum of pth absolute powers of increments of a process. This gives new insight in the use of quadratic and realised power variation as an estimate for the integrated volatility in finance. It also provides a criterion to decide from high frequency data, whether a jump component should be included in the model. Furthermore, results on the asymptotic behaviour of integrals with respect to Levy processes, estimates for integrals with respect to Levy measures and nonparametric estimation for Levy processes will be derived and viewed in the framework of variational sums.
Properties of realized variance for a pure jump process: Calendar time sampling versus business time sampling
, 2004
"... Comments are welcome In this paper we study the impact of market microstructure effects on the properties of realized variance using a pure jump process for high frequency security prices. Closed form expressions for the bias and mean squared error of realized variance are derived under alternative ..."
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Cited by 14 (0 self)
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Comments are welcome In this paper we study the impact of market microstructure effects on the properties of realized variance using a pure jump process for high frequency security prices. Closed form expressions for the bias and mean squared error of realized variance are derived under alternative sampling schemes. Importantly, we show that business time sampling is generally superior to the common practice of calendar time sampling in that it leads to a reduction in mean squared error. Using IBM transaction data we estimate the model parameters and determine the optimal sampling frequency for each day in the data set. The empirical results reveal a downward trend in optimal sampling frequency over the last 4 years with considerable daytoday variation that is closely related to changes in market liquidity.
Modelling Realized Variance when Returns are Serially Correlated
, 2002
"... This article examines the impact of serial correlation in high frequency returns on the realized variance measure. In particular, it is shown that the realized variance measure yields a biased estimate of the conditional return variance when returns are serially correlated. Using 10 years of FTSE10 ..."
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Cited by 11 (0 self)
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This article examines the impact of serial correlation in high frequency returns on the realized variance measure. In particular, it is shown that the realized variance measure yields a biased estimate of the conditional return variance when returns are serially correlated. Using 10 years of FTSE100 minute by minute data we demonstrate that a careful choice of sampling frequency is crucial in avoiding substantial biases. Moreover, we find that the autocovariances of returns disappears under temporal aggregation at a rate of decay that is consistent with an ARMA process under temporal aggregation. A simple autocovariance function based method is proposed for choosing the “optimal ” sampling frequency, that is, the highest available frequency at which the serial correlation of returns has a negligible impact on the realized variance measure. We find that the logarithmic realized variance series of the FTSE100 index, constructed using an optimal sampling frequency of 25 minutes, can be modelled as an ARFIMA process. Exogenous variables such as lagged returns and contemporaneous trading volume appear to be highly significant regressors and are able to explain a large portion of the variation in daily realized variance.
2004), “A Discrete Sine Transform Approach for Realized Volatility Measurement,” Working
, 2004
"... Realized volatility affords the expost empirical measurement of the latent notional volatility. However, the timevarying returns autocorrelation induced by microstructure effects represents a challenging problem for standard volatility measures. In this study, a new nonparametric volatility measur ..."
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Cited by 10 (0 self)
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Realized volatility affords the expost empirical measurement of the latent notional volatility. However, the timevarying returns autocorrelation induced by microstructure effects represents a challenging problem for standard volatility measures. In this study, a new nonparametric volatility measures based on the Discrete Sine Transform (DST) is proposed. We show that the DST exactly diagonalizes the covariance matrix of MA(1) process. This original result provides us an orthonomal basis decomposition of the return process which permits to optimally disentangle the underlying efficient price signal from the timevarying nuisance component contained in tickbytick return series. As a result, two nonparametric volatility estimators which fully exploit all the available information contained in high frequency data are constructed. Moreover the DST orthogonalization allow us to analytically compute the score and the Fischer information matrix of MA(1) processes. In discussing efficient numerical procedures for the likelihood maximizations we also suggest that DST estimator would represent the most valid starting point for the numerical maximization of the likelihood. Monte Carlo simulations based on a realistic model for microstructure effects show the superiority of DST estimators, compared to alternative local volatility proxies for every level of the noise to signal ratio and a large class of noise contaminations. These properties make the DST approach a nonparametric method able to cope with timevarying autocorrelation, in a simple and efficient way, providing robust and accurate volatility estimates under a wide set of realistic conditions. Moreover, its computational efficiency makes it well suitable for realtime analysis of high frequency data.
Estimation of Integrated Volatility in Stochastic Volatility Models
 tk ∈ R+ such that X(t1) = · · · = X(tk) = x. If k = 2 (or 3), then
, 2005
"... In the framework of stochastic volatility models we examine estimators for the integrated volatility based on the pth power variation, i.e. the sum of pth absolute powers of the logreturns. We derive consistency and distributional results for the estimators given high frequency data, especial ..."
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Cited by 7 (0 self)
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In the framework of stochastic volatility models we examine estimators for the integrated volatility based on the pth power variation, i.e. the sum of pth absolute powers of the logreturns. We derive consistency and distributional results for the estimators given high frequency data, especially taking into account what kind of process we may add to our model without e#ecting the estimate of the integrated volatility. This may on the one hand be interpreted as a possible flexibility in modelling, e.g. adding jumps or even leaving the framework of semimartingales by adding a fractional Brownian motion, or on the other hand as robustness against model misspecification.
A fourier transform method for nonparametric estimation of multivariate volatility
 Annals of Statistics
"... We provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semimartingales, which is based on Fourier analysis. The covolatility is reconstructed as a stochastic function of time by establishing a connection between the Fourier transform of the pr ..."
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Cited by 7 (0 self)
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We provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semimartingales, which is based on Fourier analysis. The covolatility is reconstructed as a stochastic function of time by establishing a connection between the Fourier transform of the prices process and the Fourier transform of the covolatility process. A nonparametric estimator is derived given a discrete unevenly spaced and asynchronously sampled observations of the asset price processes. The asymptotic properties of the random estimator are studied: namely, consistency in probability uniformly in time and convergence in law to a mixture of Gaussian distributions. 1. Introduction. The
Robustness of Fourier Estimator of Integrated Volatility in the Presence of Microstructure Noise
"... We study the finite sample properties of the Fourier estimator of integrated volatility under market microstructure noise. We derive an analytic expression for the bias and the mean squared error of the contaminated estimator. These estimates can be practically used to design optimal MSEbased estim ..."
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Cited by 6 (1 self)
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We study the finite sample properties of the Fourier estimator of integrated volatility under market microstructure noise. We derive an analytic expression for the bias and the mean squared error of the contaminated estimator. These estimates can be practically used to design optimal MSEbased estimators, which are very robust and efficient in the presence of noise. Moreover an empirical analysis based on a simulation study and on highfrequency logarithmic prices of the Italian stock index futures (FIB30) validates the theoretical results. JEL: C10,C13,C14,C15,C22