Realized Volatility Research Articles

Power Variations and Testing for Co‐Jumps: The Small Noise Approach

AbstractIn this paper, we study the effects of noise on bipower variation, realized volatility (RV) and testing for co‐jumps in high‐frequency data under the small noise framework. We first establish asymptotic properties of bipower variation in this framework. In the presence of the small noise, RV is asymptotically biased, and the additional asymptotic conditional variance term appears in its limit distribution. We also propose consistent estimators for the asymptotic variances of RV. Second, we derive the asymptotic distribution of the test statistic proposed in (Ann. Stat. 37, 1792‐1838) under the presence of small noise for testing the presence of co‐jumps in a two‐dimensional Itô semimartingale. In contrast to the setting in (Ann. Stat. 37, 1792‐1838), we show that the additional asymptotic variance terms appear and propose consistent estimators for the asymptotic variances in order to make the test feasible. Simulation experiments show that our asymptotic results give reasonable approximations in the finite sample cases.

Do we need the constant term in the heterogenous autoregressive model for forecasting realized volatilities?

ABSTRACTNo-constant strategy is considered for the heterogenous autoregressive (HAR) model of Corsi, which is motivated by smaller biases of its estimated HAR coefficients than those of the constant HAR model. The no-constant model produces better forecasts than the constant model for four real datasets of the realized volatilities (RVs) of some major assets. Robustness of forecast improvement is verified for other functions of realized variance and log RV and for the extended datasets of all 20 RVs of Oxford-Man realized library. A Monte Carlo simulation also reveals improved forecasts for some historic HAR model estimated by Corsi.

Modeling and forecasting oil price risk: the role of implied volatility index

PurposeWhile numerous empirical studies have tried to model and forecast the oil price volatility over the years, such attempts using the crude oil volatility index (OVX) rarely exist. In order to conceal this void, the purpose of this paper is to investigate whether including OVX in the realized volatility (RV) models improve the accuracy of predictions.Design/methodology/approachAt the empirical stage, the authors employ several measures to frame the RV of crude oil futures returns. In particular, the authors use three different range-based RV estimators recommended by Parkinson (1980), Rogers and Satchell (1991) and Alizadeh et al. (2002), respectively.FindingsThe findings reveal that the information content of crude OVX helps to provide more accurate volatility predictions in comparison to the base-line RV model which contains only historical oil volatilities. Besides, the forecast encompassing test further suggests that the modified RV model (when OVX is introduced in the base-line RV model) forecast encompasses the conventional RV forecast in majority of the cases.Practical implicationsSince forecasting oil price volatility plays a vital role in portfolio optimization, derivatives pricing, optimum asset allocation decisions and risk management, the findings of this study thus carry important implications for energy economists, investors and policymakers.Originality/valueThis paper adds to the existing literature, since it is one of the initial studies to explore whether OVX is informative about the realized variance of the US oil market returns. The findings recommend that the information content of oil implied volatilities should be taken into account when modeling the US oil market volatility. In addition, range-based measures should be utilized while estimating the RV.

Forecasting realized volatility based on the truncated two-scales realized volatility estimator (TTSRV): Evidence from China's stock market

In this paper, the TTSRV (truncated two-scale realized volatility estimator), a novel estimator of the continuous part of realized volatility (RV), is used to forecast the RV of the SSEC index. Based on the classic heterogeneous autoregressive model for RV (HAR-RV), our new model, which applies the TTSRV, can describe the continuous and jump processes of RV with higher accuracy. The empirical results obtained by this study suggest that the TTSRV outperforms previous models in both statistical and economic aspects.

주성분을 이용한 다변량 고빈도 실현 변동성의 주기 선택

본 논문은 다변량 실현 변동성 계산에서 주기 선택 방안에 대해 연구하고 있다. 고빈도(high frequency) 시계열 자료에 기초한 일간 변동성인 실현변동성을 계산하고 차원 축소 방법인 주성분을 도입하였다. Cholesky 모형을 포함한 다양한 다변량 변동성모형을 주성분을 통해 비교하였으며 KOSPI/삼성전자/현대차 고빈도 수익률 자료를 이용하여 예시하였다. We investigate multivariate volatilities based on high frequency time series. The PCA (principal component analysis) method is employed to achieve a dimension reduction in multivariate volatility. Multivariate realized volatilities (RV) with various frequencies are calculated from high frequency data and "optimum" frequency is suggested using PCA. Specifically, RVs with various frequencies are compared with existing daily volatilities such as Cholesky, EWMA and BEKK after dimension reduction via PCA. An analysis of high frequency stock prices of KOSPI, Samsung Electronics and Hyundai motor company is illustrated.

Oil price uncertainty and clean energy stock returns: New evidence from crude oil volatility index

Earlier studies evidence that oil price shocks have significant impacts on clean energy stock returns. While the previous literature uses traditional oil price series to investigate such effects, we aim to assess whether the variance of alternative energy stock returns can be explained using the information content of crude oil volatility index (OVX), an indicator of oil price uncertainty. To serve our purpose, we employ several measures to frame the realized volatility (RV) of alternative energy sector equity returns. In particular, we use three different range-based RV estimators recommended by Parkinson (1980), Rogers and Satchell (1991) and Alizadeh et al. (2002) respectively. Our findings reveal that clean energy stock market returns are highly sensitive to OVX shocks. Thus oil market uncertainty, measured by OVX, embodies a crucial role in modeling the volatility of renewable energy equity returns. In addition, we find a strong indication that OVX provides additional information beyond what is contained in the historical volatilities of equity returns. We also document that the magnitude of the effect of OVX is much bigger than that of the realized variance of WTI oil spot prices. Finally, we find that the information content of crude oil volatility index improves the volatility forecasts for the clean energy equity market.

이상치에 근거한 선택적 실현변동성 예측 방법

실현변동성(RVs)이 지속적인 장기기억성과 상당히 큰 이상치의 존재로 인해 정상계열과 비정상계열의 경계에 위치한다는 것에 주목하였다. 실현변동성을 예측하기 위해 실현변동성 이상치 관측 유무에 따라 heterogeneous autoregressive (HAR) 모형과 integrated HAR (IHAR) 모형을 번갈아 사용하는 새로운 방법을 제안하였고, 이 방법을 IHAR-O-HAR라 칭하였다. 예측력 비교는 주요 지수인 S&P 500, Nasdaq과 Nikkei 225의 실현변동성 데이터를 이용하였으며 표본 외 예측력 비교에서 새로운 IHAR-O-HAR 방법은 RW 방법, HAR 방법이나 IHAR 방법의 예측력보다 우수함을 확인하였다. We note that the dynamics of realized volatilities (RVs) are near the boundary between stationarity and non-stationarity because RVs have persistent long-memory and are often subject to fairly large outlying values. To forecast realized volatility, we consider a new method that adaptively use models with and without unit root according to the abnormality of observed RV: heterogeneous autoregressive (HAR) model and the Integrated HAR (IHAR) model. The resulting method is called the IHAR-O-HAR method. In an out-of-sample forecast comparison for the realized volatility datasets of the 3 major indexes of the S&P 500, the NASDAQ, and the Nikkei 225, the new IHAR-O-HAR method is shown superior to the existing HAR and IHAR method.

On the relation between implied and realized volatility indices: Evidence from the BRIC countries

This paper investigates the relation between implied (IV) and realized volatility (RV). Using monthly data from the BRIC countries, we assess the informational content of IV in explaining future RV as well as its unbiasedness and efficiency. We employ an ADL (Autoregressive Distributed Lag) and the corresponding EC (Error Correction) model and compare the results with the ones obtained from the OLS regression. Our goal is to assess the fully dynamical relations between these variables and to separate the short from the long-run effects. We found different results for the informational content of IV according to the methodologies used. However, both methods show that IV is an unbiased estimate of RV for India and that IV was not found to be efficient in any of the BRIC countries. Further, EC results reveal the presence of short and long-run effects for India, whereas Russia exhibits only short-run adjustments.

Volatility spillover and multivariate volatility impulse response analysis of GFC news events

ABSTRACTThis article applies two measures to assess spillovers across markets: the Diebold and Yilmaz’s (2012) spillover index and the Hafner and Herwartz’s (2006) analysis of multivariate GARCH models using volatility impulse response analysis. We use two sets of data, daily realized volatility (RV) estimates taken from the Oxford-Man RV library, for the S&P500 and the FTSE, plus 10 years of daily returns series for the New York Stock Exchange Index and the FTSE 100 index. Both data sets capture both the global Financial Crisis (GFC) and the subsequent European Sovereign Debt Crisis (ESDC). The spillover index captures the transmission of volatility to and from markets, plus net spillovers. The Volatility Impulse Responses (VIRF) have to be calibrated to conditional volatility estimated at a particular point in time. We explore the impact of three different shocks, the onset of the GFC, the height of the GFC, and the impact of the ESDC. Our modelling includes leverage and asymmetric effects applying a multivariate GARCH model, and further analysis using both BEKK and diagonal BEKK (DBEKK) models. We find the impact of negative shocks is larger, but shorter in duration, in this case a difference between 3 and 6 months.

Daily value-at-risk modeling and forecast evaluation: The realized volatility approach

One of the main applications of conditional volatility modeling and forecasting of financial assets is the value-at-risk (VaR) estimation that is used by financial institutions for reporting the daily capital in risk. It remains a question on whether realized volatility (RV) models that incorporate the use of intraday data produce better VaR forecasts compared to methodologies that are based solely on daily returns. This study provides extensive comparison of out-of-sample volatility and VaR forecast performance on three equity market indices: S&P500, FTSE100, and DAX30 using 13 risk models that consist of 5 GARCH specifications, 4 ARFIMAX specifications and 4 HARX specifications. The out-of-sample volatility forecasts are evaluated by various loss functions and simple scoring procedures in order to identity the model that produces the overall best volatility forecasts. For VaR forecasts, the models are evaluated using a two-stage backtesting procedure where the models undergo unconditional and conditional coverage tests to eliminate underperforming models and the qualified models are then evaluated using the quadratic probability score (QPS) function that is computed based on various VaR loss functions. The results showed that RV models outperform GARCH models for volatility forecasts, but a simple EGARCH model outperforms the rest models for most of the VaR forecasts. The results also indicated that capturing the asymmetric behavior of volatility dynamics is essential for accurate volatility and VaR forecasts. The findings of this study provide useful information for market risk regulation, financial risk management and further investigations such as extension to derivative markets and options pricing.

An Analysis of the Performance of Genetic Programming for Realised Volatility Forecasting

Abstract Traditionally, the volatility of daily returns in financial markets is modeled autoregressively using a time-series of lagged information. These autoregressive models exploit stylised empirical properties of volatility such as strong persistence, mean reversion and asymmetric dependence on lagged returns. While these methods can produce good forecasts, the approach is in essence atheoretical as it provides no insight into the nature of the causal factors and how they affect volatility. Many plausible explanatory variables relating market conditions and volatility have been identified in various studies but despite the volume of research, we lack a clear theoretical framework that links these factors together. This setting of a theory-weak environment suggests a useful role for powerful model induction methodologies such as Genetic Programming (GP). This study forecasts one-day ahead realised volatility (RV) using a GP methodology that incorporates information on market conditions including trading volume, number of transactions, bid-ask spread, average trading duration (waiting time between trades) and implied volatility. The forecasting performance from the evolved GP models is found to be significantly better than those numbers of benchmark forecasting models drawn from the finance literature, namely, the heterogeneous autoregressive (HAR) model, the generalized autoregressive conditional heteroscedasticity (GARCH) model, and a stepwise linear regression model (SR). Given the practical importance of improved forecasting performance for realised volatility this result is of significance for practitioners in financial markets.

Volatility spillover between the Korean KOSPI and the Hong Kong HSI stock markets

We investigate volatility spillover aspects of realized volatilities (RVs) for the log returns of the Korea Composite Stock Price Index (KOSPI) and the Hang Seng Index (HSI) from 2009-2013. For all RVs, significant long memories and asymmetries are identified. For a model selection, we consider three commonly used time series models as well as three models that incorporate long memory and asymmetry. Taking into account of goodness-of-fit and forecasting ability, Leverage heteroskedastic autoregressive realized volatility (LHAR) model is selected for the given data. The LHAR model finds significant decompositions of the spillover effect from the HSI to the KOSPI into moderate negative daily spillover, positive weekly spillover and positive monthly spillover, and from the KOSPI to the HSI into substantial negative weekly spillover and positive monthly spillover. An interesting result from the analysis is that the daily volatility spillover from the HSI to the KOSPI is significant versus the insignificant daily volatility spillover of the KOSPI to HSI. The daily volatility in Hong Kong affects next day volatility in Korea but the daily volatility in Korea does not affect next day volatility in Hong Kong.

Realised Volatility Forecasts for Stock Index Futures Using the HAR Models with Bayesian Approaches *

Abstract We investigate the realised volatility (RV) forecasts for the short, mid, and long term by developing the HAR models with Bayesian approaches and employing the high-frequency data of the China Stock Index 300 (CSI300) future for the period from 16 April 2010 to 21 May 2014. We also evaluate the performances of competing models for both in-sample forecasts and out-of-sample forecasts. We find that the proposed HAR-type models with Bayesian approaches capture the time-varying properties of parameters and predictor sets. We also find that the HAR-type models with Bayesian approaches have superior forecast performance for both in-sample forecasts and out-of-sample forecasts as compared with the benchmark HAR-type models.

장외시간 수익률을 반영한 실현변동성 추정치들의 비교

본 논문은 장외거래 수익률을 이용하여 추정한 여러 실현변동성들을 실증적으로 비교분석한다. 실제 금융 자산 시장에서는 장외시간이나 휴일에 거래가 적거나 드물게 나타나기 때문에 하루 전체의 실현변동성을 정확히 계산하는데 문제가 발생한다. 이를 해결하기 위해 제안되어진 장외거래 수익률을 여러 가지 방법으로 반영한 다양한 실현변동성의 추정치들에 대한 검토가 이루어진다. 실제 데이터의 실현변동성 추정치들의 예측정확성을 비교하기 위해 미국의 NASDAQ 지수와 S&P500 지수와 우리나라의 KOSPI 지수와 원/달러환율이 분석된다. 적분변동성의 불편추정치인 다음날의 로그수익률의 제곱을 기준으로 일일 실현 변동성의 추정치들은 비교되어지며 비교를 위해 절대평균오차(MAE)와 제곱평균오차근(RMSE)이 이용된다. 또한 통계적 추론을 위하여 Model Confidence Set(MCS) 방법과 Diebold-Mariano 검정법을 사용한다. 세 가지 주가지수 데이터에 대해 동일한 최적 방법이 선택되어지는데, 장외시간 수익률을 이용하여 장내시간 실현변동성의 크기 조정을 한 방법이다. This study makes an empirical comparison of various realized volatilities (RVs) in terms of overnight returns. In financial asset markets, during overnight or holidays, no or few trading data are available causing a difficulty in computing RVs for a whole span of a day. A review will be made on several RVs reflecting overnight return variations. The comparison is made for forecast accuracies of several RVs for some financial assets: the US S&P500 index, the US NASDAQ index, the KOSPI (Korean Stock Price Index), and the foreign exchange rate of the Korea won relative to the US dollar. The RV of a day is compared with the square of the next day log-return, which is a proxy for the integrated volatility of the day. The comparison is made by investigating the Mean Absolute Error (MAE) and the Root Mean Square Error (RMSE). Statistical inference of MAE and RMSE is made by applying the model confidence set (MCS) approach and the Diebold-Mariano test. For the three index data, a specific RV emerges as the best one, which addresses overnight return variations by inflating daytime RV.

An integrated heteroscedastic autoregressive model for forecasting realized volatilities

A new strategy for forecasting realized volatility (RV) is proposed for the heteroscedastic autoregressive (HAR) model of Corsi (2009). The strategy is constraining the sum of the HAR coefficients to one, resulting in an integrated model, called IHAR model. The IHAR model is motivated by stationarity of estimated HAR model, downward biases of estimated HAR coefficients, and over-rejection of ADF test for long-memory processes. Considerable out-of-sample forecast improvements of the IHAR model over the HAR model are demonstrated for RVs of 4 financial assets: the US S&P 500 index, the US NASDAQ index, the Japan yen/US dollar exchange rate, and the EU euro/US dollar exchange rate. Forecast improvement is also verified in a Monte Carlo experiment and in an empirical comparison for an extended data set. The forecast improvement is shown to be a consequence of the fact that the IHAR model takes better advantage of the long memory of RV and the conditional heteroscedasticity of RV than the HAR model.

A Hybrid SLV Model with Multifactor Stochastic Volatility

The need for more satisfactory pricing models has brought the attention of researchers and practitioners on Stochastic Local Volatility models. Despite the growing interest on the topic, however, it seems that no particular attention has been paid to the use of multifactor specifications for the stochastic volatility part. The additional flexibility given by this kind of approach in describing the skew dynamics becomes extremely important when we deal with the pricing of forward volatility sensitive contracts like forward start or cliquet options.This paper tries to fill the gap: we introduce the Wishart Stochastic Local Volatility model that describes the stochastic volatility by means of a Wishart process as proposed. The goal of the new framework is twofold: adding a realistic volatility dynamics to Local Volatility models without the limitations of standard 1-factor Stochastic Volatility approaches, and improving European claims pricing performance of pure Wishart Stochastic Volatility model. The latter is quite relevant since we need to introduce stringent parameters restrictions to satisfy existence and uniqueness conditions for the solution of Wishart SDE. These conditions are required when we want to properly simulate the variance process in order to price more exotic derivatives. However such conditions are not usually met when market calibration is performed and so the resulting constrained parameters set is not able to reproduce accurately the market implied volatility surface. The additional Local Volatility component acts then as compensator.We further present an innovative simulation scheme for the asset path in the pure stochastic volatility specification based on the approach. The simulation scheme is also used to tackle the calibration of the so-called leverage function and price derivatives in the new hybrid framework.

On the usefulness of intraday price ranges to gauge liquidity in cap-based portfolios

We find that easy-to-observe price ranges are useful for estimating intraday liquidity. Following the literature on range-based volatility estimators, we go beyond the use of the closing price only and rely on the full range of prices. Based on high, low, opening, and closing (HLOC) prices, we show that a greater intensity in the price discovery process (as measured by the open–close range) and a higher level of price uncertainty (as captured by the High–Low range) lower ex-ante liquidity for small, mid, and large caps. Realized volatility (RV) fails to capture these effects. Although order books have become increasingly difficult to treat, there is some good news: it has never been easier to look at price ranges.

Long-memories and mean breaks in realized volatilities

An extended sequential test of Bai and Perron (1998) to a long-memory process is applied to four sets of realized volatilities (RVs) of the US dollar–EU euro, the Japan yen–US dollar, the Korea won–US dollar exchange rates and the S&P 500 index to find significant structural breaks in the means. Even after the mean breaks are adjusted out, the RVs still have persistent memories, which will be shown to produce better out-of-sample forecasts of RVs if properly addressed than ignored. Contrary to the recent report of Choi et al. (2010) that ‘short-memory + break’ models have better forecast power than ‘long-memory only’ models in forecasting some foreign exchange rate RVs, models with ‘long-memory + mean breaks’ turn out to produce better out-of-sample forecasts than models with ‘short-memory + mean breaks’ and models with ‘long-memory only’.

European Option Pricing Under Fractional Brownian Motion with an Application to Realized Volatility

This study investigates European option pricing under fractional Brownian motion (fBm) and applies it to realized volatility (RV). The RV measure is selected because it uniquely exhibits simultaneous stationarity and long-range dependency properties in financial time series, as shown in our empirical study. Meanwhile, the Black-Scholes differential equation is not well defined when the underlying assets follow fBm with the Hurst exponent H not equal to 1/2 because fBm is not a semimartingale. Thus, we compute the European option prices using a previously proposed fractional Black-Scholes formula. Our empirical study is conducted on Tokyo Stock Price Index data from January 06, 1997 to December 30, 2013 with a sample size of 4177.

A Lagrangian multiplier test for market microstructure noise with applications to sampling interval determination for realized volatilities

A Lagrangian multiplier test is proposed for testing market microstructure noise (MMN) in financial asset prices. The test is very simple and is asymptotically chi-squared with 1-degree of freedom. The test is applied to sampling interval determination for realized volatilities (RVs) which validates the commonly used “ad hoc rule of between 5 and 30 min” for sampling interval. The proposed test gives a statistical justification for RVs of negligible serial correlation in the log-returns owning to MMN for sampling interval larger than a selected one. A Monte Carlo experiment shows reasonable size and power performance of the test. The proposed test is illustrated for two real data sets.