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Debiased Kernel Estimation of Spot Volatility in the Presence of Infinite Variation Jumps

B. Cooper Boniece, José E. Figueroa-López, Tianwei Zhou

TL;DR

This work tackles spot-volatility estimation for It\^o semimartingales with infinite-variation jumps by introducing truncated kernel estimators coupled with debiasing to push efficiency beyond the classical boundary $Y<4/3$ to $0<Y<20/11$, aided by unbounded kernels that reduce asymptotic variance. The authors derive moment expansions for truncated increments, establish feasible and fully efficient CLTs under multiple bandwidth regimes, and implement a two-step debiasing scheme that achieves rate-optimality in a wide range of jump activity. They demonstrate that the debiased estimators retain robustness and attain significantly better finite-sample performance than existing methods, including those based on empirical characteristic functions, across both stable and truncated-jump models. The results offer a practically implementable, variance-efficient approach to real-time volatility estimation in markets with high jump activity, with clear guidance on kernel choice and bandwidth tuning.

Abstract

Volatility estimation is a central problem in financial econometrics, but becomes particularly challenging when jump activity is high, a phenomenon observed empirically in highly traded financial securities. In this paper, we revisit the problem of spot volatility estimation for an Itô semimartingale with jumps of unbounded variation. We construct truncated kernel-based estimators and debiased variants that extend the efficiency frontier for spot volatility estimation in terms of the jump activity index $Y$, raising the previous bound $Y<4/3$ to $Y<20/11$, thereby covering nearly the entire admissible range $Y<2$. Compared with earlier work, our approach attains smaller asymptotic variances through the use of unbounded kernels, is simpler to implement, and has broader applicability under more flexible model assumptions. A comprehensive simulation study confirms that our procedures substantially outperform competing methods in finite samples.

Debiased Kernel Estimation of Spot Volatility in the Presence of Infinite Variation Jumps

TL;DR

This work tackles spot-volatility estimation for It\^o semimartingales with infinite-variation jumps by introducing truncated kernel estimators coupled with debiasing to push efficiency beyond the classical boundary to , aided by unbounded kernels that reduce asymptotic variance. The authors derive moment expansions for truncated increments, establish feasible and fully efficient CLTs under multiple bandwidth regimes, and implement a two-step debiasing scheme that achieves rate-optimality in a wide range of jump activity. They demonstrate that the debiased estimators retain robustness and attain significantly better finite-sample performance than existing methods, including those based on empirical characteristic functions, across both stable and truncated-jump models. The results offer a practically implementable, variance-efficient approach to real-time volatility estimation in markets with high jump activity, with clear guidance on kernel choice and bandwidth tuning.

Abstract

Volatility estimation is a central problem in financial econometrics, but becomes particularly challenging when jump activity is high, a phenomenon observed empirically in highly traded financial securities. In this paper, we revisit the problem of spot volatility estimation for an Itô semimartingale with jumps of unbounded variation. We construct truncated kernel-based estimators and debiased variants that extend the efficiency frontier for spot volatility estimation in terms of the jump activity index , raising the previous bound to , thereby covering nearly the entire admissible range . Compared with earlier work, our approach attains smaller asymptotic variances through the use of unbounded kernels, is simpler to implement, and has broader applicability under more flexible model assumptions. A comprehensive simulation study confirms that our procedures substantially outperform competing methods in finite samples.
Paper Structure (15 sections, 4 theorems, 250 equations, 4 tables)

This paper contains 15 sections, 4 theorems, 250 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Setting and background
  3. Main results
  4. Simulation
  5. Simulation with Stable Jump Component
  6. Simulation with truncated stable jump component
  7. Additional technical proofs
  8. Proof of Proposition \ref{['prop']}
  9. Proof of \ref{['GNMExp']}
  10. Proof of \ref{['PartExp0']}
  11. Proof of \ref{['PartExp0_difference']}
  12. Analog of Lemma 5 in BONIECE2024
  13. Proof of \ref{['prop1_non_integer']}
  14. Proofs of some technical identities in Theorem \ref{['clt']}
  15. Proof of technical limits in Theorem \ref{['thm_clt_difference']}

Key Result

Proposition 1

Suppose that $Y \in (0,2)\setminus\{1\}$. Let $\Delta_n^{\frac{1}{2}-s}\ll v_n \ll \Delta_n^{\frac{1}{4-Y}}$ for a fixed $s\in(0,1/2)$. Then, the following statements hold:

Theorems & Definitions (18)

  • Remark 1
  • Proposition 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • proof : Proof of Theorem \ref{['clt']}
  • Theorem 2
  • Remark 4
  • proof
  • Theorem 3
  • ...and 8 more