Efficient Sampling for Realized Variance Estimation in Time-Changed Diffusion Models

TL;DR

The paper tackles efficient realized variance estimation by sampling intraday returns in intrinsic time within a tick-time stochastic volatility (TTSV) framework that separates trading intensity and tick variance. It derives finite-sample MSE decompositions showing that homogenizing intraday returns (HTS) minimizes MSE in noise-free environments, while a novel realized BTS (rBTS) scheme, which blends observed ticks with estimated tick variance, dominates when price observations are contaminated by market microstructure noise. The TTSV model provides a transparent IV decomposition, $IV(0,T)=\int_0^T \lambda(r)\varsigma^2(r)\,dr$, enabling precise comparisons across sampling schemes and linking efficiency to the information available. Extensive simulations and an empirical NYSE application corroborate the theoretical results, highlighting practical gains in estimation accuracy and forecast performance, and offering a robust alternative when MMN is present.

Abstract

This paper analyzes the benefits of sampling intraday returns in intrinsic time for the realized variance (RV) estimator. We theoretically show in finite samples that depending on the permitted sampling information, the RV estimator is most efficient under either hitting time sampling that samples whenever the price changes by a pre-determined threshold, or under the new concept of realized business time that samples according to a combination of observed trades and estimated tick variance. The analysis builds on the assumption that asset prices follow a diffusion that is time-changed with a jump process that separately models the transaction times. This provides a flexible model that allows for leverage specifications and Hawkes-type jump processes and separately captures the empirically varying trading intensity and tick variance processes, which are particularly relevant for disentangling the driving forces of the sampling schemes. Extensive simulations confirm our theoretical results and show that for low levels of noise, hitting time sampling remains superior while for increasing noise levels, realized business time becomes the empirically most efficient sampling scheme. An application to stock data provides empirical evidence for the benefits of using these intrinsic sampling schemes to construct more efficient RV estimators as well as for an improved forecast performance.

Figures (21)

• Figure 1: Illustration of the arrival and sampling times in the TTSV model: The upper panel shows the evolution of the jump process $N(t)$ generating the ticks (arrival times) $t_i$. The lower panel shows the log-price process $P(t)$, which exhibits price jumps at the ticks $t_i$ of $N(t)$ and is constant in between. The vertical red lines represent the sampling times of an exemplary sampling scheme $\boldsymbol{\tau}$ (that does not have to be equidistant in calendar time), and the red squares show the resampled prices based on the previous tick method.
• Figure 2: IBM transaction log-prices on May 1, 2015 for three minutes in the morning between 9:45am and 9:48am and in the afternoon between 15:57pm and 16:00pm. We observe a clear pattern of much more ticks in the afternoon and a much higher "tick-by-tick" variance in the morning that is typical for stocks traded at the NYSE.
• Figure 3: Estimates of the trading intensity $\lambda(t)$, tick variance $\varsigma^2(t)$ and spot variance $\sigma^2(t)$, averaged over all trading days in the year 2018. We use the nonparametric kernel estimators for $\lambda(t)$ and $\varsigma^2(t)$ of dahlhaus2016, that we augment with a "mirror image" bias correction of DiggleMarron1988, similar to oomen2006. Following Proposition \ref{['prop:vola_decomposition']}, the estimate of the spot variance $\sigma^2(t)$ is obtained as the product of the estimated $\lambda(t)$ and $\varsigma^2(t)$.
• Figure 4: IBM log-price on May 1, 2015 together with the CTS, rTTS, rBTS and HTS sampling schemes for $M=26$, i.e., corresponding to intrinsic time 15 minute returns. For the rBTS scheme, we estimate the tick variance $\varsigma^2(\cdot)$ as the average of the estimates over the past 50 days using the estimator of dahlhaus2016. For HTS, we choose the threshold $\delta=0.00158$ that happens to result in exactly 26 sampled observations on the given day.
• Figure 5: Simulated paths of the asset price as described in Section \ref{['sec:Simulation']}, the spot variance $\sigma^2(t)$, the trading intensity $\lambda(t)$, and the tick variance $\varsigma^2(t)$ for three exemplary days in green, orange and pink. The black lines show the (appropriately rescaled according to the expected behavior of the Hawkes processes) deterministic components $\lambda_\text{det}(t)$, $\varsigma^2_\text{det}(t)$ and the resulting $\sigma^2_\text{det}(t) = \lambda_\text{det}(t) \, \varsigma^2_\text{det}(t)$ of our simulation setup that are obtained as the estimates from the IBM stock averaged over all tradings days in the year 2018.

Theorems & Definitions (29)

• Proposition 1
• Proposition 2
• Proposition 3
• Theorem 4
• Theorem 5
• Corollary 6
• Corollary 7
• Theorem 8
• Remark 9
• proof : Proof of Theorem \ref{['thm:unbiasedness']}