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Volatility of Volatility and Leverage Effect from Options

Carsten H. Chong, Viktor Todorov

Abstract

We propose model-free (nonparametric) estimators of the volatility of volatility and leverage effect using high-frequency observations of short-dated options. At each point in time, we integrate available options into estimates of the conditional characteristic function of the price increment until the options' expiration and we use these estimates to recover spot volatility. Our volatility of volatility estimator is then formed from the sample variance and first-order autocovariance of the spot volatility increments, with the latter correcting for the bias in the former due to option observation errors. The leverage effect estimator is the sample covariance between price increments and the estimated volatility increments. The rate of convergence of the estimators depends on the diffusive innovations in the latent volatility process as well as on the observation error in the options with strikes in the vicinity of the current spot price. Feasible inference is developed in a way that does not require prior knowledge of the source of estimation error that is asymptotically dominating.

Volatility of Volatility and Leverage Effect from Options

Abstract

We propose model-free (nonparametric) estimators of the volatility of volatility and leverage effect using high-frequency observations of short-dated options. At each point in time, we integrate available options into estimates of the conditional characteristic function of the price increment until the options' expiration and we use these estimates to recover spot volatility. Our volatility of volatility estimator is then formed from the sample variance and first-order autocovariance of the spot volatility increments, with the latter correcting for the bias in the former due to option observation errors. The leverage effect estimator is the sample covariance between price increments and the estimated volatility increments. The rate of convergence of the estimators depends on the diffusive innovations in the latent volatility process as well as on the observation error in the options with strikes in the vicinity of the current spot price. Feasible inference is developed in a way that does not require prior knowledge of the source of estimation error that is asymptotically dominating.
Paper Structure (15 sections, 5 theorems, 85 equations, 1 figure, 4 tables)

This paper contains 15 sections, 5 theorems, 85 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Setting
  3. Option-Based Volatility Estimators
  4. Volatility of Volatility and Leverage Effect Estimators
  5. Monte Carlo Study
  6. Setup
  7. Return-Based Volatility of Volatility and Leverage Effect Estimators
  8. Results
  9. Using Alternative Volatility Proxies
  10. Empirical Illustration
  11. Data
  12. Results
  13. Conclusion
  14. High-Frequency Expansions of Volatility Estimators Based on Characteristic Functions
  15. Proofs

Key Result

Theorem 4.1

Let $F:(0,\infty)\to\mathbb{R}$ be a $C^2$-function and $\tau_n(x)=x\mathbf 1_{\{\lvert x\rvert\leq \upsilon_n\}}$. Suppose that the log-price process $x$, the volatility process ${\sigma}$, the observed option prices and the sequences $k_n$, $\Delta_n$, $T$, $\delta$ and $\upsilon_n$ satisfy Assump where and Then, for any $0<\underline t<t<\overline t<\infty$ and $u>0$, the estimators in eq:vov

Figures (1)

  • Figure 1: S&P 500 Index Volatility Risk. The top panel displays a $5$-day moving average of annualized $\sqrt{\widehat{V}_{t,T}(u)}$; the middle panel displays $5$-day moving averages of $\widehat{VV}_{t,T}$ (black line) and $\widehat{VV}_{t,T,T'}$ (red line); the bottom panel displays $5$-day moving averages of $\widehat{LV}_{t,T}$ (black line) and $\widehat{LV}_{t,T,T'}$ (red line).

Theorems & Definitions (9)

  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.3
  • Remark 4.4
  • Theorem A.1
  • Proposition B.1
  • proof
  • proof : Proof of Theorem \ref{['thm:vov']} and Theorem \ref{['thm:lev']}
  • Lemma B.2