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Asymptotic Expansions for High-Frequency Option Data

Carsten H. Chong, Viktor Todorov

TL;DR

This paper addresses inference on volatility dynamics from high-frequency option data by deriving a nonparametric, higher-order double-asymptotic expansion for the increments of the conditional characteristic function $\mathcal{L}_{t,T}(u)$. It develops a deep It\ô semimartingale framework and proves a main expansion for $\Delta^n_i \mathcal{L}_{t,T}(u)$ that isolates the contribution of the spot volatility, its time variation and the jump components. The authors then translate the expansion into practical tools: bias-reduced option-based estimators of the instantaneous variance and transformed variance $V_t=F(\sigma^2_t)$ and a nonparametric test for finite versus infinite variation volatility jumps, with both simulations and a SPX empirical application supporting infinite variation. The results facilitate precise inference on volatility dynamics from option data and provide a robust testing device against finite-variation jump models, with clear implications for volatility-of-volatility and leverage-effect analysis.

Abstract

We derive a nonparametric higher-order asymptotic expansion for small-time changes of conditional characteristic functions of Itô semimartingale increments. The asymptotics setup is of joint type: both the length of the time interval of the increment of the underlying process and the time gap between evaluating the conditional characteristic function are shrinking. The spot semimartingale characteristics of the underlying process as well as their spot semimartingale characteristics appear as leading terms in the derived asymptotic expansions. The analysis applies to a general class of Itô semimartingales that includes in particular Lévy-driven SDEs and time-changed Lévy processes. The asymptotic expansion results are subsequently used to construct a test for whether volatility jumps are of finite or infinite variation. In an application to high-frequency data of options written on the S\&P 500 index, we find evidence for infinite variation volatility jumps.

Asymptotic Expansions for High-Frequency Option Data

TL;DR

This paper addresses inference on volatility dynamics from high-frequency option data by deriving a nonparametric, higher-order double-asymptotic expansion for the increments of the conditional characteristic function . It develops a deep It\ô semimartingale framework and proves a main expansion for that isolates the contribution of the spot volatility, its time variation and the jump components. The authors then translate the expansion into practical tools: bias-reduced option-based estimators of the instantaneous variance and transformed variance and a nonparametric test for finite versus infinite variation volatility jumps, with both simulations and a SPX empirical application supporting infinite variation. The results facilitate precise inference on volatility dynamics from option data and provide a robust testing device against finite-variation jump models, with clear implications for volatility-of-volatility and leverage-effect analysis.

Abstract

We derive a nonparametric higher-order asymptotic expansion for small-time changes of conditional characteristic functions of Itô semimartingale increments. The asymptotics setup is of joint type: both the length of the time interval of the increment of the underlying process and the time gap between evaluating the conditional characteristic function are shrinking. The spot semimartingale characteristics of the underlying process as well as their spot semimartingale characteristics appear as leading terms in the derived asymptotic expansions. The analysis applies to a general class of Itô semimartingales that includes in particular Lévy-driven SDEs and time-changed Lévy processes. The asymptotic expansion results are subsequently used to construct a test for whether volatility jumps are of finite or infinite variation. In an application to high-frequency data of options written on the S\&P 500 index, we find evidence for infinite variation volatility jumps.
Paper Structure (12 sections, 9 theorems, 284 equations, 2 figures)

This paper contains 12 sections, 9 theorems, 284 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and Assumptions
  3. Deep Itô Semimartingales
  4. Assumptions for Main Theorem
  5. The Main Expansion Theorem
  6. Application to Option-based Volatility Estimators
  7. Testing whether volatility jumps have finite or infinite variation
  8. Numerical Experiments
  9. Monte Carlo Study
  10. Empirical Illustration
  11. Proofs for Sections \ref{['sec:expansion-2']} and \ref{['sec:app']}
  12. Proof of Theorem \ref{['thm:test']}

Key Result

Theorem 3.1

For $s,t,T>0$ and $u\in\mathbb{R}\setminus\{0\}$, define where Under Assumption ass:main, there is a finite number of Itô semimartingales $v^{(k)}_t$ and $C^{(k)}_t(u)$, $k=1,\dots, K$, such that $C^{(k)}_t(u)$ is uniformly bounded in $u$ on compacts and $\lvert \Delta^n_i v^{(k)}_t\rvert+\lvert \Delta^n_i C^{(k)}_t(u) \rvert= O^\mathrm{uc} (\sqrt{\Delta_n})$,

Figures (2)

  • Figure 1: Values of $R^n_{t,T,T'}(u,U)$ in the Monte Carlo. Red line corresponds to $5\%$ critical value for a one-sided test of the null hypothesis.
  • Figure 2: Values of $R^n_{t,T,T'}(u,U)$ for the S&P 500 index. Red line corresponds to $5\%$ critical value for a one-sided test of the null hypothesis.

Theorems & Definitions (21)

  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.2
  • Remark 3.3
  • Theorem 4.1
  • Corollary 4.2
  • Corollary 4.3
  • Remark 4.4
  • Theorem 5.1
  • Proposition 7.1
  • ...and 11 more