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Asymptotic theory of range-based multipower variation

Kim Christensen, Mark Podolskij

Abstract

In this paper, we present a realized range-based multipower variation theory, which can be used to estimate return variation and draw jump-robust inference about the diffusive volatility component, when a high-frequency record of asset prices is available. The standard range-statistic -- routinely used in financial economics to estimate the variance of securities prices -- is shown to be biased when the price process contains jumps. We outline how the new theory can be applied to remove this bias by constructing a hybrid range-based estimator. Our asymptotic theory also reveals that when high-frequency data are sparsely sampled, as is often done in practice due to the presence of microstructure noise, the range-based multipower variations can produce significant efficiency gains over comparable subsampled return-based estimators. The analysis is supported by a simulation study and we illustrate the practical use of our framework on some recent TAQ equity data.

Asymptotic theory of range-based multipower variation

Abstract

In this paper, we present a realized range-based multipower variation theory, which can be used to estimate return variation and draw jump-robust inference about the diffusive volatility component, when a high-frequency record of asset prices is available. The standard range-statistic -- routinely used in financial economics to estimate the variance of securities prices -- is shown to be biased when the price process contains jumps. We outline how the new theory can be applied to remove this bias by constructing a hybrid range-based estimator. Our asymptotic theory also reveals that when high-frequency data are sparsely sampled, as is often done in practice due to the presence of microstructure noise, the range-based multipower variations can produce significant efficiency gains over comparable subsampled return-based estimators. The analysis is supported by a simulation study and we illustrate the practical use of our framework on some recent TAQ equity data.
Paper Structure (20 sections, 6 theorems, 96 equations, 5 figures, 3 tables)

This paper contains 20 sections, 6 theorems, 96 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theoretical framework
  3. The model
  4. The data
  5. Return-based estimation of quadratic variation
  6. Subsampling
  7. Range-based estimation of quadratic variation
  8. Extension to jump-diffusion processes
  9. realized range-based multipower variation
  10. The joint distribution of $RRV_{b}^{n,m}$ and $RTV^{n,m}$
  11. Simulation study
  12. Simulation results
  13. Empirical application
  14. Conclusions and directions for future research
  15. Acknowledgements
  16. ...and 5 more sections

Key Result

Theorem 1

Assume that $p$ follows the jump-diffusion process defined in Eq. BSMJ. As $n \to \infty$, it holds that:

Figures (5)

  • Figure 1: Asymptotic variance factor of realized range-based estimators.
  • Figure 2: Asymptotic approximation of the realized range-based tripower variance.
  • Figure 3: Jump-robust 95% confidence intervals for the integrated variance.
  • Figure 4: Illustration using MRK.
  • Figure 5: Comparison of return- and range-based estimator of the integrated variance, SPY data.

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Definition
  • Remark
  • Theorem 2
  • proof
  • Remark
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • ...and 1 more