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Nonparametric inference for spot volatility in pure-jump semimartingales

Chengxin Yan, Dachuan Chen, Jia Li

TL;DR

This paper develops nonparametric spot volatility inference for pure-jump semimartingales driven by a symmetric $\beta$-stable process, introducing both fixed-$k$ and large-$k$ asymptotics. It provides estimators and coupling results under active and inactive jump regimes, including when $\beta$ is unknown but consistently estimable, and derives Gaussian and stable limits for the resulting t-statistics. Monte Carlo evidence shows the fixed-$k$ approach offers substantially better finite-sample accuracy than large-$k$, guiding practical application in high-frequency settings. The framework yields valid, nonparametric confidence intervals for spot volatility in jump-dominated models, with broad applicability to nonparametric volatility analysis beyond Brownian settings.

Abstract

We provide a comprehensive analysis of spot volatility inference in pure-jump semimartingales under two asymptotic settings: fixed-$k$, where each local window uses a fixed number of observations, and large-$k$, where this number grows with sampling frequency. For both active- and possibly inactive-jump settings, we derive generally nonstandard, typically non-Gaussian limit distributions and establish valid inference, including when the jump-activity index is consistently estimated. Simulations show that fixed-$k$ asymptotics offer markedly better finite-sample accuracy, underscoring their practical advantage for nonparametric spot volatility inference.

Nonparametric inference for spot volatility in pure-jump semimartingales

TL;DR

This paper develops nonparametric spot volatility inference for pure-jump semimartingales driven by a symmetric -stable process, introducing both fixed- and large- asymptotics. It provides estimators and coupling results under active and inactive jump regimes, including when is unknown but consistently estimable, and derives Gaussian and stable limits for the resulting t-statistics. Monte Carlo evidence shows the fixed- approach offers substantially better finite-sample accuracy than large-, guiding practical application in high-frequency settings. The framework yields valid, nonparametric confidence intervals for spot volatility in jump-dominated models, with broad applicability to nonparametric volatility analysis beyond Brownian settings.

Abstract

We provide a comprehensive analysis of spot volatility inference in pure-jump semimartingales under two asymptotic settings: fixed-, where each local window uses a fixed number of observations, and large-, where this number grows with sampling frequency. For both active- and possibly inactive-jump settings, we derive generally nonstandard, typically non-Gaussian limit distributions and establish valid inference, including when the jump-activity index is consistently estimated. Simulations show that fixed- asymptotics offer markedly better finite-sample accuracy, underscoring their practical advantage for nonparametric spot volatility inference.
Paper Structure (13 sections, 17 theorems, 191 equations, 1 figure)

This paper contains 13 sections, 17 theorems, 191 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fixed- inference theory
  3. Spot volatility inference with known $\beta$
  4. The case with active jumps
  5. The case with possibly inactive jumps
  6. Spot volatility inference with unknown $\beta$
  7. Large- inference theory
  8. The case with active jumps
  9. The case with possibly inactive jumps
  10. Simulations
  11. Proofs
  12. Proofs in Section \ref{['sect:fixed-k']}
  13. Proofs in Section \ref{['sect:large-k']}

Key Result

Theorem 1

Suppose Assumption assumption holds and $\beta \in (1,2)$. Then it holds that for any $t \in \mathcal{T}_{n,j}$, where $(Z_i)_{i \in \mathcal{I}_{n,j}}$ are i.i.d. $\beta$-stable random variables satisfying $\mathbb{E}[e^{iuZ_i}] = \exp(-|u|^{\beta}/2)$.

Figures (1)

  • Figure 1: Monte Carlo comparison between finite-sample distribution of $\hat{\sigma}_{n,j}(p)/\sigma_{n,t}^p$ and fixed-$k$ and large-$k$ limits.

Theorems & Definitions (32)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Example 1
  • Example 2
  • Example 3
  • Lemma 1
  • Corollary 3
  • Theorem 3
  • ...and 22 more