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In this paper, under suitable assumptions on the Gaussian process G = { G t , t ≥ 0 } , we establish results on uniform convergence in probability and in law stably for the realized power variation of the Riemann–Stieljes integral Z t = ∫ 0 t u s d Y s , G (1) with respect to Y t , G (1) = ∫ 0 t e - s d G a (s) , where u is a process of finite q-variation with q < 1 / (1 - α) , α ∈ (0 , 1) and a (t) = α e t α . To illustrate the results, we show that the required conditions on G are satisfied for processes including fractional Brownian motion with Hurst parameter α ∈ (0 , 1) , subfractional Brownian motion of index α ∈ (0 , 1 / 2) and bifractional Brownian motion of parameters (α , K) ∈ (0 , 1 / 2) × (0 , 1 ] . Furthermore, we apply our results to construct an estimator for the integrated volatility parameter in an Ornstein–Uhlenbeck model driven by Y G (1) . [ABSTRACT FROM AUTHOR]

Rounding is a pivotal source of market microstructure noise which should be carefully addressed in high-frequency data analysis. This study incorporates available market information in modeling rounding errors and proposes a Rounding Error-Trading Information model from the Bayesian perspective. We assign a thresholded Gaussian process prior to the instantaneous volatility of the log-price process and adopt a fully Bayesian approach with an efficient Markov chain Monte Carlo algorithm for model inference, based on which a novel Trading Information-based estimator for the integrated volatility is provided. Simulation studies show that the proposed method can effectively recover the true latent log-prices, instantaneous volatility, and integrated volatility with multiple volatility shapes and rounding mechanisms, even under model misspecification. Extensive empirical studies show that the rounding mechanism in the Shanghai A-share market is likely to be random and the trading directions have a greater impact on the rounding results of the asset prices than the true latent log-prices, which is a consistent finding with that in the literature. [ABSTRACT FROM AUTHOR]

We focus on estimating daily integrated volatility (IV) by realized measures based on intraday returns following a discrete-time stochastic model with a pronounced intraday periodicity (IP). We demonstrate that neglecting the IP-impact on realized estimators may lead to invalid statistical inference concerning IV for a common finite number of intraday returns. For a given IP functional form, we analytically derive robust IP-correction factors for realized measures of IV as well as their asymptotic distributions. We show both in Monte Carlo simulations and empirically that the proposed bias corrections are the robust way to account for IP by computing realized estimators. [ABSTRACT FROM AUTHOR]

In this paper, under suitable assumptions on the Gaussian process G = { G t , t ≥ 0 } , we establish results on uniform convergence in probability and in law stably for the realized power variation of the Riemann–Stieljes integral Z t = ∫ 0 t u s d Y s , G (1) with respect to Y t , G (1) = ∫ 0 t e - s d G a (s) , where u is a process of finite q-variation with q < 1 / (1 - α) , α ∈ (0 , 1) and a (t) = α e t α . To illustrate the results, we show that the required conditions on G are satisfied for processes including fractional Brownian motion with Hurst parameter α ∈ (0 , 1) , subfractional Brownian motion of index α ∈ (0 , 1 / 2) and bifractional Brownian motion of parameters (α , K) ∈ (0 , 1 / 2) × (0 , 1 ] . Furthermore, we apply our results to construct an estimator for the integrated volatility parameter in an Ornstein–Uhlenbeck model driven by Y G (1) . [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of the estimation of the integrated volatility of log-prices based on high frequency data when both price jumps and market microstructure noise are present. We begin by providing a survey of the leading estimators introduced in the literature to tackle volatility estimation in this setting. We then introduce novel integrated volatility estimators based on a truncation technique and establish their properties. Finally, we carry out a simulation study to compare the performance of the different estimation techniques. [ABSTRACT FROM AUTHOR]

We give a general time-varying parameter model, where the multidimensional parameter possibly includes jumps. The quantity of interest is defined as the integrated value over time of the parameter process Θ = T − 1 ∫ 0 T θ t * d t . We provide a local parametric estimator (LPE) of Θ and conditions under which we can show the central limit theorem. Roughly speaking those conditions correspond to some uniform limit theory in the parametric version of the problem. The framework is restricted to the specific convergence rate n1∕2. Several examples of LPE are studied: estimation of volatility, powers of volatility, volatility when incorporating trading information and time-varying MA(1). [ABSTRACT FROM AUTHOR]

In this paper, we examine whether jumps matter in both equity market returns and integrated volatility. For this purpose, we use the swap variance (SwV) approach to identify monthly jumps and estimated realized volatility in prices for both developed and emerging markets from February 2001 to February 2020. We find that jumps arise in all equity markets; however, emerging markets have more jumps relative to developed markets, and positive jumps are more frequent than negative jumps. In emerging markets, the markets with average volatility earn higher returns during jump periods; however, highly volatile markets earn higher returns during jump periods in developed markets. Furthermore, markets with low continuous returns and high volatility are more adversely affected during periods of negative jumps. The average ratio of jump variations to total variation shows considerable variations due to jumps. Integrated volatility is high during periods of negative jumps, and this pattern is consistent in both developed and emerging markets. Moreover, the peak volatility of stock markets is observed during periods of crises. The implication of this study is useful in the asset pricing model, risk management, and for individual investors and portfolio managers for both developed and emerging markets.

This paper introduces a solution that combines the Kalman and particle filters to the challenging problem of estimating integrated volatility using high‐frequency data where the underlying prices are perturbed by a mixture of random noise and price discreteness. An explanation is presented of how the proposed combined filtering approach is able to correct for bias due to this mixed‐type microstructure effect. Simulation and empirical studies on the tick‐by‐tick trade price data for four US stocks in the year 2009 show that our method has clear advantages over existing high‐frequency volatility estimation methods. [ABSTRACT FROM AUTHOR]

Higlights • Finding a consistent and asymptotically unbiased estimator of integrated volatility for emerging markets under consideration. • Determining optimal slow frequency for two time scale estimation. • Employing the upper tail dependence (Gumbel copula) measure for comparison purposes, in addition to standard loss functions MSE and QLIKE. • Employing DM test in determining forecasting accuracy between competing OHLC estimators. • Recommending the appropriate ex-post daily volatility measure in the lack of high-frequency data for each emerging market under consideration. Abstract This paper explores the effectiveness of eight range-based volatility estimators for seven European emerging markets. It offers added value by: (i) finding a consistent and asymptotically unbiased estimator of integrated volatility for emerging markets, (ii) employing the upper tail dependence for comparison purposes, in addition to standard loss functions, and (iii) recommending the appropriate ex-post volatility measure in the lack of high-frequency data. When no strong preference for a specific estimator is found, the upper tail dependence measure is consulted, confirming the MSE-based ranking for Czech Republic, Greece, Poland, and Romania; and the QLIKE-based ranking for Bulgaria, Croatia, and Hungary. [ABSTRACT FROM AUTHOR]

We investigate the marginal predictive content of small versus large jump variation, when forecasting one-week-ahead cross-sectional equity returns, building on Bollerslev et al. (2020). We find that sorting on signed small jump variation leads to greater value-weighted return differentials between stocks in our highest- and lowest-quintile portfolios (i.e., high–low spreads) than when either signed total jump or signed large jump variation is sorted on. It is shown that the benefit of signed small jump variation investing is driven by stock selection within an industry, rather than industry bets. Investors prefer stocks with a high probability of having positive jumps, but they also tend to overweight safer industries. Also, consistent with the findings in Scaillet et al. (2018), upside (downside) jump variation negatively (positively) predicts future returns. However, signed (large/small/total) jump variation has stronger predictive power than both upside and downside jump variation. One reason large and small (signed) jump variation have differing marginal predictive contents is that the predictive content of signed large jump variation is negligible when controlling for either signed total jump variation or realized skewness. By contrast, signed small jump variation has unique information for predicting future returns, even when controlling for these variables. By analyzing earnings announcement surprises, we find that large jumps are closely associated with “big” news. However, while such news-related information is embedded in large jump variation, the information is generally short-lived, and dissipates too quickly to provide marginal predictive content for subsequent weekly returns. Finally, we find that small jumps are more likely to be diversified away than large jumps and tend to be more closely associated with idiosyncratic risks. This indicates that small jumps are more likely to be driven by liquidity conditions and trading activity.

The models of the GARCH family, normally used for the estimates of volatility for longer periods, keep unchanged the relative weights assigned to the observations both old and new, regardless of the volatility´s forecasted horizon. The purpose of this article is to verify if the increase in relative weights assigned to the earlier observations due to the increase of the forecast horizon results in better estimates of volatility. Through the use of seven forecasting models of volatility and return series of financial markets assets, the estimates obtained in the sample (in-sample) were compared with observations outside the sample (out-of-sample). Based on this comparison, it was found that the best estimates of expected volatility were obtained by the modified EGARCH model and the ARLS model. We conclude that the use of traditional forecasting models of volatility, which keep unchanged relative weights assigned to both old and new observations, was inappropriate.

This article examines the leverage effect, or the generally negative covariation between asset returns and their changes in volatility, under a general setup that allows the log-price and volatility processes to be Itô semimartingales. We decompose the leverage effect into continuous and discontinuous parts and develop statistical methods to estimate them. We establish the asymptotic properties of these estimators. We also extend our methods and results (for the continuous leverage) to the situation where there is market microstructure noise in the observed returns. We show in Monte Carlo simulations that our estimators have good finite sample performance. When applying our methods to real data, our empirical results provide convincing evidence of the presence of the two leverage effects, especially the discontinuous one. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

We propose a new method to implement the Business Time Sampling (BTS) scheme for high-frequency financial data. We compute a time-transformation (TT) function using the intraday integrated volatility estimated by a jump-robust method. The BTS transactions are obtained using the inverse of the TT function. Using our sampled BTS transactions, we test the semi-martingale hypothesis of the stock log-price process and estimate the daily realized volatility. Our method improves the normality approximation of the standardized business-time return distribution. Our Monte Carlo results show that the integrated volatility estimates using our proposed sampling strategy provide smaller root mean-squared error. [ABSTRACT FROM AUTHOR]

The integrated volatility plays an important role in risk management and portfolio selection, the estimation methods regarding the quantity have been widely investigated, either under low-frequency data or high-frequency data, or a combination of both. In this paper, we propose a measure for the integrated volatility via limit order book data with possible presence of multiple records. The estimator is valid under mild conditions and it is easily implemented. The finite sample performance of the proposed estimator has been verified by simulation studies and we apply the method to some real high-frequency data-sets as well. [ABSTRACT FROM PUBLISHER]

In this paper, we develop the multipower estimators for the integrated volatility in (Barndorff-Nielsen and Shephard in J. Financ. Econom. 2:1-37, 2004); these estimators allow the presence of jumps in the underlying driving process and the simultaneous presence of microstructure noise and multiple records of observations. By multiple records we mean more than one observation recorded on a single time stamp, as often seen in stock markets, in particular, for heavily traded securities, for a data set with even millisecond frequency. We establish the consistency and asymptotic normality of the estimators for both noise-free and noise-present cases. Simulation studies confirm our theoretical results. We apply the estimators to a real high-frequency data set. [ABSTRACT FROM AUTHOR]

This paper investigates the eigenspace estimation problem for the large integrated volatility matrix based on non-synchronized and noisy observations from a high-dimensional Itô process. We establish a minimax lower bound for the eigenspace estimation problem and propose sparse principal subspace estimation methods by using the multi-scale realized volatility matrix estimator or the pre-averaging realized volatility matrix estimator. We derive convergence rates of the proposed eigenspace estimators and show that the estimators can achieve the minimax lower bound, and thus are rate-optimal. The minimax lower bound can be established by Fano’s lemma with an appropriately constructed subclass that has independent but not identically distributed normal random variables with zero mean and heterogeneous variances. [ABSTRACT FROM AUTHOR]

In financial practices and research studies, we often encounter a large number of assets. The availability of high-frequency financial data makes it possible to estimate the large volatility matrix of these assets. Existing volatility matrix estimators such as kernel realized volatility and pre-averaging realized volatility perform poorly when the number of assets is very large, and in fact they are inconsistent when the number of assets and sample size go to infinity. In this paper, we introduce threshold rules to regularize kernel realized volatility, pre-averaging realized volatility, and multi-scale realized volatility. We establish asymptotic theory for these threshold estimators in the framework that allows the number of assets and sample size to go to infinity. Their convergence rates are derived under sparsity on the large integrated volatility matrix. In particular we have shown that the threshold kernel realized volatility and threshold pre-averaging realized volatility can achieve the optimal rate with respect to the sample size through proper bias corrections, but the bias adjustments cause the estimators to lose positive semi-definiteness; on the other hand, in order to be positive semi-definite, the threshold kernel realized volatility and threshold pre-averaging realized volatility have slower convergence rates with respect to the sample size. A simulation study is conducted to check the finite sample performances of the proposed threshold estimators with over hundred assets. [ABSTRACT FROM AUTHOR]

This article provides a data-driven analysis of the volatility risk premium, using tools from high-frequency finance and Big Data analytics. We argue that the volatility risk premium, loosely defined as the difference between realized and implied volatility, can best be understood when viewed as a systematically priced bias. We first use ultra-high-frequency transaction data on SPDRs and a novel approach for estimating integrated volatility on the frequency domain to compute realized volatility. From that we subtract the daily VIX, our measure of implied volatility, to construct a time series of the volatility risk premium. To identify the factors behind the volatility risk premium as a priced bias, we decompose it into magnitude and direction. We find compelling evidence that the magnitude of the deviation of the realized volatility from implied volatility represents supply and demand imbalances in the market for hedging tail risk. It is difficult to conclusively accept the hypothesis that the direction or sign of the volatility risk premium reflects expectations about future levels of volatility. However, evidence supports the hypothesis that the sign of the volatility risk premium is indicative of gains or losses on a delta-hedged portfolio. [ABSTRACT FROM PUBLISHER]

This paper develops a method to improve the estimation of jump variation using high frequency data with the existence of market microstructure noises. Accurate estimation of jump variation is in high demand, as it is an important component of volatility in finance for portfolio allocation, derivative pricing and risk management. The method has a two-step procedure with detection and estimation. In Step 1, we detect the jump locations by performing wavelet transformation on the observed noisy price processes. Since wavelet coefficients are significantly larger at the jump locations than the others, we calibrate the wavelet coefficients through a threshold and declare jump points if the absolute wavelet coefficients exceed the threshold. In Step 2 we estimate the jump variation by averaging noisy price processes at each side of a declared jump point and then taking the difference between the two averages of the jump point. Specifically, for each jump location detected in Step 1, we get two averages from the observed noisy price processes, one before the detected jump location and one after it, and then take their difference to estimate the jump variation. Theoretically, we show that the two-step procedure based on average realized volatility processes can achieve a convergence rate close to OP(n-4/9) which is better than the convergence rate OP(n-1/4) for the procedure based on the original noisy process, where n is the sample size. Numerically, the method based on average realized volatility processes indeed performs better than that based on the price processes. Empirically, we study the distribution of jump variation using Dow Jones Industrial Average stocks and compare the results using the original price process and the average realized volatility processes. [ABSTRACT FROM AUTHOR]

Estimation of the integrated volatility is an important problem in high-frequency financial data analysis. In this study, we propose a quadratic unbiased estimator of the integrated volatility for stochastic volatility models with microstructure noise. The proposed estimator minimizes the finite sample variance in the class of quadratic estimators based on symmetric Toeplitz matrices. We show the proposed estimator has an asymptotic mixed normal distribution with optimal convergence rate $$n^{-1/4}$$ and achieves the maximum likelihood estimator efficiency for constant volatility case. Simulation results show that our proposed estimator attains better finite sample efficiency than state-of-the-art methods. Finally, a real data analysis is conducted for illustration. [ABSTRACT FROM AUTHOR]

Accurate volatility forecasting is a key determinant for portfolio management, risk management and economic policy. The paper provides evidence that the sum of squared standardized forecast errors is a reliable measure for model evaluation when the predicted variable is the intra-day realized volatility. The forecasting evaluation is valid for standardized forecast errors with leptokurtic distribution as well as with leptokurtic and asymmetric distributions. Additionally, the widely applied forecasting evaluation function, the predicted mean-squared error, fails to select the adequate model in the case of models with residuals that are leptokurtically and asymmetrically distributed. Hence, the realized volatility forecasting evaluation should be based on the standardized forecast errors instead of their unstandardized version. [ABSTRACT FROM AUTHOR]

This article studies the effect of market microstructure noise on volatility estimation in the frequency domain. We propose a bias-corrected periodogram-based estimator of integrated volatility. We show that the new estimator is consistent and the central limit theorem is established under a general assumption of the noise. We also provide a feasible procedure for computing the bias-corrected estimator in practice. As a byproduct, we extract a consistent frequency-domain estimator of the long-run variance of market microstructure noise from high-frequency data. [ABSTRACT FROM AUTHOR]

We consider noisy non-synchronous discrete observations of a continuous semimartingale with random volatility. Functional stable central limit theorems are established under high-frequency asymptotics in three setups: one-dimensional for the spectral estimator of integrated volatility, from two-dimensional asynchronous observations for a bivariate spectral covolatility estimator and multivariate for a local method of moments. The results demonstrate that local adaptivity and smoothing noise dilution in the Fourier domain facilitate substantial efficiency gains compared to previous approaches. In particular, the derived asymptotic variances coincide with the benchmarks of semiparametric Cramér–Rao lower bounds and the considered estimators are thus asymptotically efficient in idealized sub-experiments. Feasible central limit theorems allowing for confidence bounds are provided. [ABSTRACT FROM AUTHOR]

We develop a framework for estimating the quadratic variation of discontinuous semi-martingales with intra-day high-low statistics. Restricting the realized rangebased variance smaller than a suitably defined threshold, we propose an integrated volatility estimator and consider its consistency and asymptotic normality under a set of weak conditions. We find that the precision of our statistics is about five times greater than that of realized variance purely restricted by threshold. Simulation results illustrate the good finite sample properties of our estimator. [ABSTRACT FROM AUTHOR]

We design adaptive realized kernels to estimate the integrated volatility in a framework that combines a stochastic volatility model with leverage effect for the efficient price and a semiparametric microstructure noise model specified at the highest frequency. Some time dependence parameters of the noise model must be estimated before adaptive realized kernels can be implemented. We study their performance by simulation and illustrate their use with twelve stocks listed in the Dow Jones Industrial. As expected, we find that adaptive realized kernels achieves the optimal trade-off between the discretization error and the microstructure noise. [ABSTRACT FROM AUTHOR]

Applying jump-robust methods to estimating integrated volatility is in the mainstream of financial econometrics. However, little if any attention has been devoted to the construction of a jump-free estimator for integrated covariance that overlooks the well-documented manifestation of joint jumps. Joint jumps are contemporaneous within the day. Therefore, this study proposes a three-point approach that not only deals with estimating volatility, but also constructs a singular-jump-free and joint-jump-free covariance. Since the basic idea of the three-point covariance is based on conditional quantiles, we also provide two alternative procedures for finding approximated estimations in practical applications. Based on this approach, our empirical results confirm that singular jumps and joint jumps occur on the Taiwan Futures Exchange. [ABSTRACT FROM PUBLISHER]

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Abstract: In this article we consider the volatility inference in the presence of both market microstructure noise and endogenous time. Estimators of the integrated volatility in such a setting are proposed, and their asymptotic properties are studied. Our proposed estimator is compared with the existing popular volatility estimators via numerical studies. The results show that our estimator can have substantially better performance when time endogeneity exists. [Copyright &y& Elsevier]

This paper elaborates sequential procedures for monitoring the validity of a volatility model. A state-space representation describes dynamics of daily integrated volatility. The observation equation relates the integrated volatility to its measures such as the realized volatility or bipower variation. On-line control procedures, based on volatility forecasting errors, allow us to decide whether the chosen representation remains correctly specified. A signal indicates that the assumed volatility model may no longer be valid. The performance of our approach is analyzed within a Monte Carlo simulation study and illustrated in an empirical application for selected U.S. stocks. [ABSTRACT FROM PUBLISHER]

By fractional integration of a square root volatility process, we propose in this paper a long memory extension of the Heston (Rev Financ Stud 6:327-343, 1993) option pricing model. Long memory in the volatility process allows us to explain some option pricing puzzles as steep volatility smiles in long term options and co-movements between implied and realized volatility. Moreover, we take advantage of the analytical tractability of affine diffusion models to clearly disentangle long term components and short term variations in the term structure of volatility smiles. In addition, we provide a recursive algorithm of discretization of fractional integrals in order to be able to implement a method of moments based estimation procedure from the high frequency observation of realized volatilities. [ABSTRACT FROM AUTHOR]

Let σ t be the instantaneous cross-volatility of two continuous semimartingales X and Y. In this paper, we introduce some estimators for the class of integrated cross-volatilities of the form where g is a continuous function and processes X and Y are sampled with microstructure noise and in an asynchronous way. In finance, it is widely accepted that the processes X and Y are reasonable models for the log return of price processes of stock and currency and our estimator is relevant in the context of intra-day high-frequency trading. [ABSTRACT FROM PUBLISHER]

Recent advances in financial econometrics have allowed for the construction of efficient ex post measures of daily volatility. This paper investigates the importance of instability in models of realised volatility and their corresponding forecasts. Testing for model instability is conducted with a subsampling method. We show that removing structurally unstable data of a short duration has a negligible impact on the accuracy of conditional mean forecasts of volatility. In contrast, it does provide a substantial improvement in a model's forecast density of volatility. In addition, the forecasting performance improves, often dramatically, when we evaluate models on structurally stable data. [ABSTRACT FROM AUTHOR]

We consider a stochastic process driven by diffusions and jumps. Given a discrete record of observations, we devise a technique for identifying the times when jumps larger than a suitably defined threshold occurred. This allows us to determine a consistent non-parametric estimator of the integrated volatility when the infinite activity jump component is Lévy. Jump size estimation and central limit results are proved in the case of finite activity jumps. Some simulations illustrate the applicability of the methodology in finite samples and its superiority on the multipower variations especially when it is not possible to use high frequency data. [ABSTRACT FROM AUTHOR]

This paper evaluates the performance of conditional variance models using high-frequency data of the National Stock Index (S&P CNX NIFTY) and attempts to determine the optimal sampling frequency for the best daily volatility forecast. A linear combination of the realized volatilities calculated at two different frequencies is used as benchmark to evaluate the volatility forecasting ability of the conditional variance models (GARCH (1, 1)) at different sampling frequencies. From the analysis, it is found that sampling at 30 minutes gives the best forecast for daily volatility. The forecasting ability of these models is deteriorated, however, by the non-normal property of mean adjusted returns, which is an assumption in conditional variance models. Nevertheless, the optimum frequency remained the same even in the case of different models (EGARCH and PARCH) and different error distribution (generalized error distribution, GED) where the error is reduced to a certain extent by incorporating the asymmetric effect on volatility. Our analysis also suggests that GARCH models with GED innovations or EGRACH and PARCH models would give better estimates of volatility with lower forecast error estimates. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The robust feed-back control schemes to provide the sustainable growth of investor capital under the absence of certain risks are introduced. These schemes are based on the current dynamics of the asset prices. It is assumed that the price of asset follows rather general stochastic differential equation. In contrast to the generally used self-financing strategy the control is realized within the framework of an open system. The latter implies the possibility to invest cash into the portfolio in the process of trading. [ABSTRACT FROM AUTHOR]

Estimation of integrated multivariate volatilities of an Itô process is an interesting and important issue in finance, for example, in order to evaluate portfolios. New non-parametric estimators have been recently proposed by Malliavin and Mancino (2002) and Hayashi and Yoshida (2005a) as alternative methods to classical realized quadratic covariation. The purpose of this article is to compare these alternative estimators both theoretically and empirically, when high frequency data is available. We found that the Hayashi-Yoshida estimator performs the best among the alternatives in view of the bias and the MSE. The other estimators are shown to have possibly heavy bias mostly toward the origin. We also applied these estimators to Japanese Government Bond futures to obtain the results consistent with our simulation. [ABSTRACT FROM AUTHOR]

The wide availability of high-frequency data for many financial instruments stimulates an upsurge interest in statistical research on the estimation of volatility. Jump-diffusion processes observed with market microstructure noise are frequently used to model high-frequency financial data. Yet existing methods are developed for either noisy data from a continuous-diffusion price model or data from a jump-diffusion price model without noise. We propose methods to cope with both jumps in the price and market microstructure noise in the observed data. These methods allow us to estimate both integrated volatility and jump variation from the data sampled from jump-diffusion price processes, contaminated with the market microstructure noise. Our approach is to first remove jumps from the data and then apply noise-resistant methods to estimate the integrated volatility. The asymptotic analysis and the simulation study reveal that the proposed wavelet methods can successfully remove the jumps in the price processes and the integrated volatility can be estimated as accurately as in the case with no presence of jumps in the price processes. In addition, they have outstanding statistical efficiency. The methods are illustrated by applications to two high-frequency exchange rate data sets. [ABSTRACT FROM AUTHOR]

We study the parametric problem of estimating the drift coefficient in a stochastic volatility model $Y_{t}=\int_{0}^{t}\sqrt{V_{s}}\,\mathrm {d}W_{s}$ , where Y is a log price process and V the volatility process. Assuming that one can recover the volatility, precisely enough, from the observation of the price process, we construct an efficient estimator for the drift parameter of the diffusion V. As an application we present the efficient estimation based on the discrete sampling $(Y_{i\delta_{n}})_{i=0,\dots,n}$ with δ n →0 and n δ n →∞. We show that our setup is general enough to cover the case of ‘microstructure noise’ for the price process as well. [ABSTRACT FROM AUTHOR]

Based on the concept of multipower variation we establish a class of easily computable and robust estimators for the integrated volatility, especially including the squared integrated volatility, in Lévy-type stochastic volatility models. We derive consistency and feasible distributional results for the estimators. Furthermore, we discuss the applications to time-changed CGMY, normal inverse Gaussian, and hyperbolic models with and without leverage, where the time-changes are based on integrated Cox—Ingersoll—Ross or Ornstein—Uhlenbeck-type processes. We deduce which type of market microstructure does not affect the estimates. [ABSTRACT FROM AUTHOR]