Nonparametric realized volatility estimation in the international equity markets

Using high-frequency intraday data, we construct, test and model seven new realized volatility estimators for six international equity indices. We detect jumps in these estimators, construct the jump components of volatility and perform various tests on their properties. Then we use the class of heterogeneous autoregressive (HAR) models for assessing the relevant effects of jumps on volatility. Our results expand and complement the previous literature on the nonparametric realized volatility estimation in terms of volatility jumps being examined and modeled for the international equity market, using such a variety of new realized volatility estimators. The selection of realized volatility estimator greatly affects jump detection, magnitude and modeling. The properties each volatility estimator tries to incorporate affect the detection, magnitude and properties of jumps. These volatility-estimation and jump properties are also evident in jump modeling based on statistical and economic terms.     

Dynamics of realized volatilities and correlations: An empirical study

This study examines two important issues underlying realized volatility and correlation estimators. First, an empirical inquiry is conducted to assess whether Bax and Eurodollar futures tick-by-tick data can be characterized as marked-point processes. Second, ARMA, neural network, GARCH-BEKK, and naive volatility and correlation forecasts are compared in an out-of-sample context when a trader prices an interest rate spread option based on those forecasts and simultaneously delta-hedges her position. Other loss functions are also considered. Competing volatility forecasts are also compared to implied volatilities.     

Finite sample accuracy and choice of sampling frequency in integrated volatility estimation

We consider the properties of three estimation methods for integrated volatility, i.e. realized volatility, Fourier, and wavelet estimation, when a typical sample of high-frequency data is observed. We employ several different generating mechanisms for the instantaneous volatility process, e.g. Ornstein–Uhlenbeck, long memory, and jump processes. The possibility of market microstructure contamination is also entertained using models with bid-ask bounce and price discreteness, in which case alternative estimators with theoretical justification under market microstructure noise are also examined. The estimation methods are compared in a simulation study which reveals a general robustness towards persistence or jumps in the latent stochastic volatility process. However, bid-ask bounce effects render realized volatility and especially the wavelet estimator less useful in practice, whereas the Fourier method remains useful and is superior to the other two estimators in that case. More strikingly, even compared to bias correction methods for microstructure noise, the Fourier method is superior with respect to RMSE while having only slightly higher bias. A brief empirical illustration with high-frequency GE data is also included.     

Portfolio analysis of intraday covariance matrix in the Greek equity market

The intraday nonparametric estimation of the variance–covariance matrix adds to the literature in portfolio analysis of the Greek equity market. This paper examines the economic value of various realized volatility and covariance estimators under the strategy of volatility timing. I use three types of portfolios: Global Minimum Variance, Capital Market Line and Capital Market Line with only positive weights. The estimators of volatilities and covariances use 5-min high-frequency intraday data. The dataset concerns the FTSE/ATHEX Large Cap index, FTSE/ATHEX Mid Cap index, and the FTSE/ATHEX Small Cap index of the Greek equity market (Athens Stock Exchange). As far as I know, this is the first work of its kind for the Greek equity market. Results concern not only the comparison of various estimators but also the comparison of different types of portfolios, in the strategy of volatility timing. The economic value of the contemporary non-parametric realized volatility estimators is more significant than this when the covariance is estimated by the daily squared returns. Moreover, the economic value (in b.p.s) of each estimator changes with the volatility timing.     

News-based sentiment and bitcoin volatility

In this work, I studied whether news media sentiments have an impact on Bitcoin volatility. In doing so, I applied three different range-based volatility estimates along with two different sentiments, namely psychological sentiments and financial sentiments, incorporating four various sentiment dictionaries. By analyzing 17,490 news coverages by 91 major English-language newspapers listed in the LexisNexis database from around the globe from January 2012 until August 2021, I found news media sentiments to play a significant role in Bitcoin volatility. Following the heterogeneous autoregressive model for realized volatility (HAR-RV)—which uses the heterogeneous market idea to create a simple additive volatility model at different scales to learn which factor is influencing the time series—along with news sentiments as explanatory variables, showed a better fit and higher forecasting accuracy. Furthermore, I also found that psychological sentiments have medium-term and financial sentiments have long-term effects on Bitcoin volatility. Moreover, the National Research Council Emotion Lexicon showed the main emotional drivers of Bitcoin volatility to be anticipation and trust.     

Range-Based Estimation of Stochastic Volatility Models

We propose using the price range in the estimation of stochastic volatility models. We show theoretically, numerically, and empirically that range-based volatility proxies are not only highly efficient, but also approximately Gaussian and robust to microstructure noise. Hence range-based Gaussian quasi-maximum 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 two-factor models with one highly persistent factor and one quickly mean-reverting factor.     

Two-step estimation of the volatility functions in diffusion models with empirical applications

In this article, we develop a two-step estimation procedure for the volatility function in diffusion models. We firstly estimate the volatility series at sampling time points based on high-frequency data. Then, the volatility function estimator can be obtained by using the kernel smoothing method. The resulting estimators are presented based on high-frequency data, and are shown to be consistent and asymptotically normal. We also consider boundary issues and then propose two methods to handle them. The asymptotic normality of two boundary-corrected estimators is established under some suitable conditions. The proposed estimators are illustrated by Monte Carlo simulations and real data.     

Implications of market microstructure for realized variance measurement

Volatility measuring and estimation based on intra-day high-frequency data has grown in popularity during the last few years. A significant part of the research uses volatility and variance measures based on the sum of squared high-frequency returns. These volatility measures, introduced and mathematically justified in a series of papers by Andersen et al. [1999. (Understanding, optimizing, using and forecasting) realized volatility and correlation. Leonard N. Stern School Finance Department Working Paper Series, 99-061, New York University; 2000a. The distribution of realized exchange rate volatility. Journal of the American Statistical Association 96, no. 453: 42–55; 2000b. Exchange rate returns standardized by realized volatility are (nearly) Gaussian. Multinational Finance Journal 4, no. 3/4: 159–179; 2003. Modeling and forecasting realized volatility. NBER Working Paper Series 8160.] and Andersen et al. 2001a. Modeling and forecasting realized volatility. NBER Working Paper Series 8160., are referred to as ‘realized variance’. From the theory of quadratic variations of diffusions, it is possible to show that realized variance measures, based on sufficiently frequently sampled returns, are error-free volatility estimates. Our objective here is to examine realized variance measures, where well-documented market microstructure effects, such as return autocorrelation and volatility clustering, are included in the return generating process. Our findings are that the use of squared returns as a measure for realized variance will lead to estimation errors on sampling frequencies adopted in the literature. In the case of return autocorrelation, there will be systematic biases. Further, we establish increased standard deviation in the error between measured and real variance as sampling frequency decreases and when volatility is non-constant.     

The economic value of range-based covariance between stock and bond returns with dynamic copulas

The covariance between stock and bond returns plays important roles in the setting up of asset allocation strategies and portfolio diversification. In the present study, we propose a multivariate range-based volatility model incorporating dynamic copulas into a range-based volatility model to describe the volatility and dependence structures of stock and bond returns. We then go on to assess the economic value of the covariance forecasts based on our proposed model under a mean-variance framework. The out-of-sample forecasting performance reveals that investors would be willing to pay between 39 and 2081 basis points per year to switch from a dynamic trading strategy under the return-based volatility model to a dynamic trading strategy under the range-based volatility model, with more risk-averse investors being willing to pay even higher switching fees. Furthermore, additional economic gains of between 33 and 1471 annualized basis points are achieved when taking the leverage effect into consideration.     

Intra-day realized volatility for European and USA stock indices

The paper constructs measures of intra-day realized volatility for 17 European and USA stock indices. We utilize a model-free de-noising method by assembling the realized volatility in sampling frequency selected according to the volatility signature plot, which minimizes the microstructure effects. Having verified the stylized facts of realized volatility, the dynamic behavior of correlation between realized volatilities is investigated. The correlation among realized volatilities is positive and extremely high, although for some periods, it decreases dramatically. The correlation of volatilities within USA (or Europe) is much higher than the correlation of volatilities across USA and Europe. Moreover, we provide evidence that the inter-day adjusted realized volatility reduces significantly the underestimation of the true variability.     

A cyclical model of exchange rate volatility

In this paper, we investigate the long run dynamics of the intraday range of the GBP/USD, JPY/USD and CHF/USD exchange rates. We use a non-parametric filter to extract the low frequency component of the intraday range, and model the cyclical deviation of the range from the long run trend as a stationary autoregressive process. We use the cyclical volatility model to generate out-of-sample forecasts of exchange rate volatility for horizons of up to 1 year under the assumption that the long run trend is fully persistent. As a benchmark, we compare the forecasts of the cyclical volatility model with those of the range-based EGARCH and FIEGARCH models of Brandt and Jones (2006). Not only does the cyclical volatility model provide a very substantial computational advantage over the EGARCH and FIEGARCH models, but it also offers an improvement in out-of-sample forecast performance.     

Frequency of observation and the estimation of integrated volatility in deep and liquid financial markets

Using two newly available ultrahigh-frequency datasets, we investigate empirically how frequently one can sample certain foreign exchange and U.S. Treasury security returns without contaminating estimates of their integrated volatility with market microstructure noise. Using the standard realized volatility estimator, we find that one can sample dollar/euro returns as frequently as once every 15 to 20 s without contaminating estimates of integrated volatility; 10-year Treasury note returns may be sampled as frequently as once every 2 to 3 min on days without U.S. macroeconomic announcements, and as frequently as once every 40 s on announcement days. Using a simple realized kernel estimator, this sampling frequency can be increased to once every 2 to 5 s for dollar/euro returns and to about once every 30 to 40 s for T-note returns. These sampling frequencies, especially in the case of dollar/euro returns, are much higher than those that are generally recommended in the empirical literature on realized volatility in equity markets. The higher sampling frequencies for dollar/euro and T-note returns likely reflect the superior depth and liquidity of these markets.     

Modeling and forecasting the additive bias corrected extreme value volatility estimator

In this paper, we provide a framework to model and forecast daily volatility based on the newly proposed additive bias corrected extreme value volatility estimator (the Add RS estimator). The theoretical framework of the additive bias corrected extreme value volatility estimator is based on the closed form solution for the joint probability of the running maximum and the terminal value of the random walk. Using the opening, high, low and closing prices of S&P 500, CAC 40, IBOVESPA and S&P CNX Nifty indices, we find that the logarithm of the Add RS estimator is approximately Gaussian and that a simple linear Gaussian long memory model can be applied to forecast the logarithm of the Add RS estimator. The forecast evaluation analysis indicates that the conditional Add RS estimator provides better forecasts of realized volatility than alternative range-based and return-based models.     

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

This article documents the conditional and unconditional distributions of the realized volatility for the 2008 futures contract in the European climate exchange (ECX), which is valid under the EU emissions trading scheme (EU ETS). Realized volatility measures from naive, kernel-based and subsampling estimators are used to obtain inferences about the distributional and dynamic properties of the ECX emissions futures volatility. The distribution of the daily realized volatility in logarithmic form is shown to be close to normal. The mixture-of-normals hypothesis is strongly rejected, as the returns standardized using daily measures of volatility clearly departs from normality. A simplified HAR-RV model (Corsi in J Financ Econ 7:174–196, 2009) with only a weekly component, which reproduces long memory properties of the series, is then used to model the volatility dynamics. Finally, the predictive accuracy of the HAR-RV model is tested against GARCH specifications using one-step-ahead forecasts, which confirms the HAR-RV superior ability.     

Central limit theorem for the realized volatility based on tick time sampling

A central limit theorem for the realized volatility estimator of the integrated volatility based on a specific random sampling scheme is proved, where prices are sampled with every ‘continued price change’ in bid or ask quotation data. The estimator is shown to be robust to market microstructure noise induced by price discreteness and bid–ask spreads. More general sampling schemes also are treated in case that the price process is a diffusion.     

Conditional risk and performance evaluation: Volatility timing, overconditioning, and new estimates of momentum alphas

Unconditional alphas are biased when conditional beta covaries with the market risk premium (market timing) or volatility (volatility timing). We demonstrate an additional bias (overconditioning) that can occur any time an empiricist estimates risk using information, such as a realized beta, that is not available to investors ex ante. Calibrating to U.S. equity returns, volatility timing and overconditioning can plausibly impact alphas more than market timing, which has been the focus of prior literature. To correct market- and volatility-timing biases without overconditioning, we show that incorporating realized betas into instrumental variables estimators is effective. Empirically, instrumentation reduces momentum alphas by 20-40%. Overconditioned alphas overstate performance by up to 2.5 times. We explain the sources of both the volatility-timing and overconditioning biases in momentum portfolios.     

The one-trading-day-ahead forecast errors of intra-day realized volatility

Two volatility forecasting evaluation measures are considered; the squared one-day-ahead forecast error and its standardized version. The mean squared forecast error is the widely accepted evaluation function for the realized volatility forecasting accuracy. Additionally, we explore the forecasting accuracy based on the squared distance of the forecast error standardized with its volatility. The statistical properties of the forecast errors point the standardized version as a more appropriate metric for evaluating volatility forecasts.We highlight the importance of standardizing the forecast errors with their volatility. The predictive accuracy of the models is investigated for the FTSE100, DAX30 and CAC40 European stock indices and the exchange rates of Euro to British Pound, US Dollar and Japanese Yen. Additionally, a trading strategy defined by the standardized forecast errors provides higher returns compared to the strategy based on the simple forecast errors. The exploration of forecast errors is paving the way for rethinking the evaluation of ultra-high frequency realized volatility models.     

Forecasting time-varying covariance with a range-based dynamic conditional correlation model

This paper proposes a range-based dynamic conditional correlation (DCC) model combined by the return-based DCC model and the conditional autoregressive range (CARR) model. The substantial gain in efficiency of volatility estimation can boost the accuracy for estimating time-varying covariances. As to the empirical study, we use the S&P 500 stock index and the 10-year treasury bond futures to examine both in-sample and out-of-sample results for six models, including MA100, EWMA_, CCC_, BEKK, return-based DCC, and range-based DCC. Of all the models considered, the range-based DCC model is largely supported in estimating and forecasting the covariance matrices.     

ABC of SV: Limited information likelihood inference in stochastic volatility jump-diffusion models

We develop novel methods for estimation and filtering of continuous-time models with stochastic volatility and jumps using so-called Approximate Bayesian Computation which build likelihoods based on limited information. The proposed estimators and filters are computationally attractive relative to standard likelihood-based versions since they rely on low-dimensional auxiliary statistics and so avoid computation of high-dimensional integrals. Despite their computational simplicity, we find that estimators and filters perform well in practice and lead to precise estimates of model parameters and latent variables. We show how the methods can incorporate intra-daily information to improve on the estimation and filtering. In particular, the availability of realized volatility measures help us in learning about parameters and latent states. The method is employed in the estimation of a flexible stochastic volatility model for the dynamics of the S&P 500 equity index. We find evidence of the presence of a dynamic jump rate and in favor of a structural break in parameters at the time of the recent financial crisis. We find evidence that possible measurement error in log price is small and has little effect on parameter estimates. Smoothing shows that, recently, volatility and the jump rate have returned to the low levels of 2004–2006.     

Foreign exchange,fractional cointegration and the implied–realized volatility relation

Almost all relevant literature has characterized implied volatility as a biased predictor of realized volatility. In this paper we provide new time series techniques to investigate the validity of this finding in several foreign exchange options markets, including the Euro market. First, we develop a new fractional cointegration test that is shown to be robust to both stationary and non-stationary regions. Second, we employ both intra-day and daily data to measure realized volatility in order to assess the relevance of data frequency in resolving the bias. Third, we use data on implied volatility traded on the market. In contrast to previous studies, we show that the frequency of data used for measuring realized volatility within a fractionally cointegrating framework is important for the results of unbiasedness tests. Significantly, for many popular exchange rates, the use of intra-day rather than daily data affects the emergence of a different bias, as the possibility of a fractionally integrated risk premium admits itself!