Estimating stochastic volatility models using daily returns and realized volatility simultaneously
Introduction
The financial return volatility, defined as the variance or the standard deviation of returns, plays a central role in modern finance such as the option pricing and the evaluation of risk measures, e.g. value-at-risk (VaR) and expected shortfall. Realized volatility, which is the sum of squared intraday returns over a certain interval such as a day, has recently attracted the attention of financial economists and econometricians as an accurate measure of the true volatility. The realized volatility, proposed by Andersen and Bollerslev (1998) and Barndorff-Nielsen and Shephard (2001) independently, would provide a consistent estimator of the latent volatility in the ideal market assumption.
In the real market, however, there are two problems in measuring daily realized volatility from high frequency return data. One problem is the presence of non-trading hours and the other is the presence of the market microstructure noise in transaction prices. Stock markets are open only for a part of a day. For example, the Tokyo Stock Exchange (TSE) is open only for 4.5 h a day. The realized volatility may underestimate the latent one-day volatility if we define the latent one-day volatility for day as the volatility from the market closing time for day to that for day as usual and calculate the realized volatility as the sum of squared intraday returns only when the market is open. To avoid this underestimation, Hansen and Lunde (2005) scale realized volatility using daily returns so that the mean of the realized volatility equals the variance of the daily return.
On the other hand, the market microstructure noise has various sources, including discrete trading and bid-ask spread (see e.g., O’Hara (1995) and Hasbrouck (2007) for details). Due to the noise, the realized volatility can be a biased estimator of the latent volatility (see, e.g., McAleer and Medeiros (2008) for a review of the realized volatility and effects of the microstructure noise). As the time interval approaches zero, the variance of the true price process independent of the market microstructure noise decreases and then the effect of the microstructure noise becomes more significant. This means that there is a trade-off between the variance and bias of the realized volatility. Considering this trade-off, Bandi and Russell (2008) derive a simple formula to produce the optimal time interval of intraday returns used for calculating the realized volatility. Zhang et al. (2005) also propose the way to correct the bias by combining two realized volatilities calculated from returns with different frequencies.
While the intraday returns are heavily contaminated by the microstructure noise, the daily returns are less subject to the noise. Thus the daily returns may provide additional information on the latent volatility. From this perspective, this article models the daily returns and realized volatility simultaneously by extending stochastic volatility models with or without asymmetry between today’s daily return and tomorrow’s volatility.
We assume that the realized volatility includes the microstructure noise but still contains much information on the latent volatility. On the other hand, daily returns have less such noises but do not include the sufficient information on the latent volatility. Therefore, the model can correct the bias using all the available high frequency data. This feature is shared basically only by the two-scale estimator of Zhang et al. (2005) while all other volatility estimators are inefficient in the sense that they discard a large amount of available data for correcting the bias. Additionally, the model can estimate the biases due to both the microstructure noise and non-trading hours simultaneously without an additional calculation to determine the optimal time interval using the formula of Bandi and Russell (2008), to compute several realized volatilities for calculating the two-scale estimator of Zhang et al. (2005), or to scale realized volatility as in Hansen and Lunde (2005). Further, modeling returns and realized volatility simultaneously has a certain advantage in that our model enables us to estimate the entire conditional predictive distribution of returns and hence common risk measures such as VaR and expected shortfall can be easily estimated.
However, it is difficult to evaluate the likelihood of our model analytically and hence to estimate the parameters in the model by the maximum likelihood method. Thus we develop a Bayesian method for estimating the parameters in our model using the Markov chain Monte Carlo (MCMC) technique. To make the estimation method efficient, we extend the block (multi-move) samplers proposed by Shephard and Pitt (1997) and Watanabe and Omori (2004) for symmetric stochastic volatility models and by Omori and Watanabe (2008) for asymmetric ones. The MCMC method also enables us to take account of the parameter uncertainty in predicting the distribution of returns.
We illustrate our model and estimation method by applying them to the daily data on returns and realized volatility of the Tokyo stock price index (TOPIX). We show that this model can estimate realized volatility biases and parameters simultaneously. Bayesian comparison between the simultaneous models using both two (naive and scaled) realized volatilities shows that the effect of non-trading hours is more essential than that of microstructure noise. Further, extending the simultaneous models with asymmetry improves the model fitting subsutantially.
The paper is organized as follows. In Section 2, we first describe how to compute the realized volatility and discuss two practical problems in such computations. Then we propose a simultaneous model and explain its estimation method using the MCMC technique. Section 3 applies our proposed model to the TOPIX data. Finally, Section 4 concludes the paper.
Section snippets
Integrated volatility, realized volatility, and microstructure noise
We first consider a simple continuous time process, where denotes the log price of a financial asset at time , and is the instantaneous or spot volatility which is assumed to have locally square integrable sample paths and be stochastically independent of the standard Brownian motion . Then, the volatility for day is defined as the integral of over the interval where a full twenty-four-hour day is represented by the time interval 1, i.e.,
Data and realized volatility
We use the high frequency data of Tokyo stock price index (TOPIX) obtained from the Nikkei NEEDS MT tick data during the period from April 1, 1996 to March 31, 2005 (2216 trading days). For this period, the highest frequency at which the price is preserved is one minute. TSE is open for 9:00–11:00 (morning session) and 12:30–15:00 (afternoon session) in usual trading days and only for 9:00–11:00 in the first and last trading days in every year. Excluding the overnight and lunch time intervals,
Concluding remarks
In this paper, we proposed modeling daily returns and realized volatility simultaneously extending the well-known stochastic volatility model and described the efficient sampling algorithm for our model to implement Markov chain Monte Carlo simulation. We show that this model can jointly estimate the parameters and the realized volatility bias due to both non-trading hours and the market microstructure noise. Especially, this model allows us to use the realized volatility calculated from all
Acknowledgements
The authors thank Herman van Dijk, Gianni Amisano, Philippe Deschamps, Erricos John Kontoghiorghes, and two anonymous referees for their helpful comments. This work is partially supported by Grants-in-Aid for Scientific Research 18330039 and for Special Purposes 18203901 from the Japanese Ministry of Education, Science, Sports, Culture and Technology, and Ishii Memorial Securities Research Promotion Foundation.
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