This chapter proposes the class of renewal based volatility estimator for high fre- quency volatility estimation, and develops its asymptotic theory of the estimator based on renewal theory. The renewal based volatility estimator differs from RV-type estimators as it does not require an equidistant deterministic sampling grid and does not rely on computing squared returns. Our theory opens up a wide range of possibilities to construct alternative volatility estimators such as range duration-based RBV-type estimators with more efficiency compared to RV-type estimators, while providing consistency and asymptotic distribution for the entire class of renewal based volatility estimators. Moreover, the stochastic sampling duration in calendar time is allowed to be parametrized, which can potentially lead to a significant efficiency gain compared to non-parametric renewal based volatility estimators.
Using the theory of renewal based volatility estimators, we prove theoretically the consistency and provide the asymptotic distribution for the point process based volatility estimator as in Engle and Russell (1998), Gerhard and Hautsch (2002), Tse and Yang (2012), Nolte, Taylor, and Zhao (2018) and Li, Nolte, and Nolte (2018b) under a continuous martingale setting. We examine Nolte, Taylor, and Zhao’s (2018) NPD estimator in detail, showing its robustness to drifts and jumps, and establishing its bias structure under MMS noise, time discretization and price discretization. In our simulation study we show that: (1) it is suboptimal to choose a very small δ due to truncation bias. (2) When the MMS noise level is small, the NPD estimator is more efficient than the calendar time estimators. (3) The NPD estimator in general is more robust to jumps than the RBip estimator. (4) The NPD estimator is much more sensitive to the level of noise compared to the calendar time methods. (5) Exponentially smoothing the contaminated price process can yield an approximately unbiased NPDz estimator that provides high efficiency compared to optimized RK and pre-averaged estimators while preserving the robustness to jumps.
This chapter has several limitations that provide rooms for future research. Firstly, the idea of a range duration-based volatility estimator can be further developed as it is showing some very promising properties under the pure diffusion assumption. Different from the realized range estimator proposed by Christensen and Podolskij (2007), the normalizing coefficient for the NPR estimator is just 0.5, and the asymp-
totic properties follow directly from our theory. However, the properties of this estimator under various noise structures are yet to be verified, but it is promising that its properties can be analysed following the same approach for the NPD estimator presented in this paper. Secondly, the properties of the PRBV estimator require
1.8 Concluding Remarks | 49
further analysis, as we assume that the renewal reward processRiis known. Therefore it is also helpful to examine the impact of estimation noise of Ri on the efficiency of the PRBV estimator. Finally, theoretical properties of the NPDz estimator and a data-driven method to select the optimal smoothing parameterγ are also worth separate investigation.
Chapter 2
High-Frequency Volatility
Estimation and the Relative
Importance of Market
Microstructure Variables
2.1
Introduction
Volatility is an important topic in financial econometrics and a crucial input for any asset pricing, portfolio allocation and risk management framework. It is considered as a latent process that describes the variability of the return process over a given in- terval, and thus requires estimation from the observed price process. The availability of high-frequency financial data has led to a shift of volatility estimators from low frequency volatility models such as GARCH-type models (Bollerslev, 1986; Engle, 1982) to high-frequency volatility measures, with the realized volatility (RV)-type estimators (Andersen, Bollerslev, Diebold, and Labys, 2001) being the most represen- tative and widely applied high-frequency volatility measures. The RV-type volatility estimators have the advantages of well-established mathematical properties and can be modified to provide precise volatility measures that are robust to various market frictions. However, this type of estimators suffer from the problem of heavy reliance on the availability of data and its non-parametric design also restricts the user to only use price information for volatility estimation.
Restricting ourselves to only price data results in large information loss for intraday volatility estimation, due to the fact that information about volatility contained in observable market microstructure (MMS) variables is completely ignored. To overcome this problem, we apply the duration-based volatility estimator initially
proposed by Engle and Russell (1998) and subsequently developed by Tse and Yang (2012), Nolte, Taylor, and Zhao (2018) and Li, Nolte, and Nolte (2018a). This estimator provides a flexible parametric structure to incorporate other explanatory variables and also produces precise volatility estimates on both daily and intraday levels.
In this chapter, we consider the following MMS variables: trading volume, bid- ask spread, total quote depth, quote depth difference, number of trades, order flow and order imbalance1. By coupling a variant of the Autoregressive Conditional Dura- tion (ACD) model (Engle and Russell, 1998) with the best subset selection regression, we conduct a novel analysis of the relative importance of the MMS variables based on their contribution to volatility estimation and provide guidance on the optimal selection of MMS variables for volatility estimation, followed by a study of the effect of inclusion of MMS variables on the quality of volatility estimates.
Our main empirical findings based on 29 highly liquid securities and a market index ETF (SPY) suggest that, firstly, order flow and number of trades possess the most important information for volatility estimation, followed by total quote depth, quote depth difference, bid-ask spread, order imbalance and trading volume. Both the ranking and the optimal choice of MMS covariates vary considerably across stocks. More importantly, we demonstrate that, by benchmarking on a realized kernel (RK) estimator, the volatility estimates obtained from the ACD model with the inclusion of optimally selected MMS covariates significantly outperform those obtained from an ACD model without any MMS covariates on both daily and intraday levels. Moreover, including all MMS covariates does not further improve the results. These findings have two important implications: (1) our variable selection procedure can to a great extent extract the most relevant information for volatility estimation; (2) MMS covariates indeed contain invaluable information about the latent price volatility process and should not be overlooked.
The contribution of this chapter is three-folded. Firstly, we develop a framework for ranking and choosing various MMS variables for high-frequency volatility estimation. Secondly, we are among the first to provide rankings and optimal choices of a universe of MMS variables based on their relative importance for volatility estimation, which provides new insights into the relationship between intraday volatility and other variables. Thirdly, we demonstrate that it is possible to obtain more precise volatility estimates by including MMS covariates, which is an important alternative to the