Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps

Abstract

We propose a new concept of modulated bipower variation for diffusion models with microstructure noise. We show that this method provides simple estimates for such important quantities as integrated volatility or integrated quarticity. Under mild conditions the consistency of modulated bipower variation is proven. Under further assumptions we prove stable convergence of our estimates with the optimal rate n −1/4 . Moreover, we construct estimates which are robust to finite activity jumps. 1 0 σ 4 s ds.This quantity is of particular interest because, properly scaled, it occurs as the conditional variance in central limit theorems for estimators of IV and has therefore to be estimated.

The Realized Empirical Distribution Function of Stochastic Variance with Application to Goodness-of-Fit Testing

“…We draw 500 bootstrap samples in each simulation trial and inspect both the setting where v is known and estimated. We follow Corradi and Distaso (2006) and recover the parameters viav using GMM to match the sample mean, variance, plus the first and second autocovariance of the bias-corrected pre-averaged bipower variation of Podolskij and Vetter (2009a) to the corresponding model-based moments of integrated variance (the former is consistent for the latter).…”

“…The table reports descriptive statistics of our equity high-frequency data. "code" is the ticker symbol, "effective sample" is the fraction of previous-tick interpolated 10-second prices originating from a new transaction rather than a repetition (a measure of liquidity), σ is annualized volatility based on a pre-averaged bipower variation of Podolskij and Vetter (2009a), while γ is the noise-tosignal ratio defined in the main text. These numbers are computed daily and averaged over the sample.…”

“…This is opposite to the estimation of integrated variance, where realized variance-either noiserobust or not-converges to the quadratic variation, which includes the sum of the squared jumps in X, and we need to enforce some robustification (e.g., truncation or bipower variation) to recover the continuous variation of X (e.g., Barndorff-Nielsen and Shephard, 2004;Jing, Liu, and Kong, 2014;Mancini, 2009;Podolskij and Vetter, 2009a). As an aside, we note Li, Todorov, and Tauchen (2013) only prove consistency of the jump-truncated statistic in the presence of discontinuous X and no microstructure noise, but this can also be extended to the non-truncated estimator.…”

Bootstrapping Pre-Averaged Realized Volatility Under Market Microstructure Noise

“…This has motivated the development of alternative estimators. One popular method is the pre-averaging approach first introduced by Podolskij and Vetter (2009) and further studied by Jacod et al (2009)…”

Inference for Continuous Semimartingales Observed at High Frequency

“…A different kind of blocking, pre-averaging, is used by Podolskij and Vetter (2006) and Jacod, Li, Mykland, Podolskij, and Vetter (2008) in the context of inference in the presence of microstructure noise. In these papers, the (latent) semimartingale is itself given a locally constant approximation.…”