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Realised quantile-based estimation of the integrated variance

Kim Christensen, Roel Oomen, Mark Podolskij

TL;DR

The paper addresses the challenge of estimating the integrated variance $IV=\int_0^1 \sigma_t^2 dt$ from high-frequency data contaminated by market microstructure noise and jumps. It introduces the quantile-based realised variance (QRV), which uses blockwise return quantiles to achieve consistency and high efficiency while remaining robust to finite-activity jumps; it then extends QRV to noisy data via a pre-averaging based estimator $QRV^*$ with a feasible CLT at the optimal rate $N^{-1/4}$. The authors develop both blocked and subsampled implementations, derive optimal quantile weights to minimize asymptotic variance, and show that efficiency can approach the ML bound as multiple quantiles are used. Through extensive simulations and empirical exercises on DJIA and Apple data, QRV demonstrates strong jump/outlier robustness and competitive finite-sample performance relative to existing estimators like RV, BPV, TRV, and MedRV, with practical guidelines for tuning and implementation. The work offers a practical, theoretically grounded tool for nonparametric volatility estimation and has significant implications for derivative pricing, risk management, and high-frequency econometrics.

Abstract

In this paper, we propose a new jump robust quantile-based realised variance measure of ex-post return variation that can be computed using potentially noisy data. The estimator is consistent for the integrated variance and we present feasible central limit theorems which show that it converges at the best attainable rate and has excellent efficiency. Asymptotically, the quantile-based realised variance is immune to finite activity jumps and outliers in the price series, while in modified form the estimator is applicable with market microstructure noise and therefore operational on high-frequency data. Simulations show that it has superior robustness properties in finite sample, while an empirical application illustrates its use on equity data.

Realised quantile-based estimation of the integrated variance

TL;DR

The paper addresses the challenge of estimating the integrated variance from high-frequency data contaminated by market microstructure noise and jumps. It introduces the quantile-based realised variance (QRV), which uses blockwise return quantiles to achieve consistency and high efficiency while remaining robust to finite-activity jumps; it then extends QRV to noisy data via a pre-averaging based estimator with a feasible CLT at the optimal rate . The authors develop both blocked and subsampled implementations, derive optimal quantile weights to minimize asymptotic variance, and show that efficiency can approach the ML bound as multiple quantiles are used. Through extensive simulations and empirical exercises on DJIA and Apple data, QRV demonstrates strong jump/outlier robustness and competitive finite-sample performance relative to existing estimators like RV, BPV, TRV, and MedRV, with practical guidelines for tuning and implementation. The work offers a practical, theoretically grounded tool for nonparametric volatility estimation and has significant implications for derivative pricing, risk management, and high-frequency econometrics.

Abstract

In this paper, we propose a new jump robust quantile-based realised variance measure of ex-post return variation that can be computed using potentially noisy data. The estimator is consistent for the integrated variance and we present feasible central limit theorems which show that it converges at the best attainable rate and has excellent efficiency. Asymptotically, the quantile-based realised variance is immune to finite activity jumps and outliers in the price series, while in modified form the estimator is applicable with market microstructure noise and therefore operational on high-frequency data. Simulations show that it has superior robustness properties in finite sample, while an empirical application illustrates its use on equity data.
Paper Structure (18 sections, 9 theorems, 75 equations, 7 figures, 2 tables)

This paper contains 18 sections, 9 theorems, 75 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Quantile-based realised variance measurement
  3. The estimator and its properties
  4. A subsampling implementation of QRV
  5. QRV with $m \to \infty$
  6. Implementation and asymptotic efficiency of QRV
  7. Finite sample performance and jump robustness
  8. QRV with market microstructure noise
  9. Construction of the estimator
  10. Asymptotic properties
  11. Finite sample performance and noise robustness
  12. Empirical illustration
  13. QRV with "clean" low-frequency data
  14. QRV with "noisy" high-frequency data
  15. Concluding remarks
  16. ...and 3 more sections

Key Result

Theorem 1

For the process $X$ in Eq. Eqn:X, and $N = mn$ with $m$ fixed, as $N \to \infty$ where $IV = \int_{0}^{1} \sigma_{u}^{2} \text{\upshape{d}}u$.

Figures (7)

  • Figure 1: Optimal quantile weights and scaling factors for varying block size $m$.
  • Figure 2: SV and jump simulation.
  • Figure 3: Performance of QRV$^\ast$ in the presence of noise.
  • Figure 4: QRV with daily Dow Jones Industrial Average index data.
  • Figure 5: Summary statistics of "noisy" AAPL trade data.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Theorem 3
  • Proposition 2
  • Remark 1
  • Theorem 4
  • Definition 2
  • Theorem 5
  • ...and 2 more