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On the estimation of leverage effect and volatility of volatility in the presence of jumps

Qiang Liu, Zhi Liu, Wang Zhou

Abstract

We study the estimation of leverage effect and volatility of volatility by using high-frequency data with the presence of jumps. We first construct spot volatility estimator by using the empirical characteristic function of the high-frequency increments to deal with the effect of jumps, based on which the estimators of leverage effect and volatility of volatility are proposed. Compared with existing estimators, our method is valid under more general jumps, making it a better alternative for empirical applications. Under some mild conditions, the asymptotic normality of the estimators is established and consistent estimators of the limiting variances are proposed based on the estimation of volatility functionals. We conduct extensive simulation study to verify the theoretical results. The results demonstrate that our estimators have relative better performance than the existing ones, especially when the jump is of infinite variation. Besides, we apply our estimators to a real high-frequency dataset, which reveals nonzero leverage effect and volatility of volatility in the market.

On the estimation of leverage effect and volatility of volatility in the presence of jumps

Abstract

We study the estimation of leverage effect and volatility of volatility by using high-frequency data with the presence of jumps. We first construct spot volatility estimator by using the empirical characteristic function of the high-frequency increments to deal with the effect of jumps, based on which the estimators of leverage effect and volatility of volatility are proposed. Compared with existing estimators, our method is valid under more general jumps, making it a better alternative for empirical applications. Under some mild conditions, the asymptotic normality of the estimators is established and consistent estimators of the limiting variances are proposed based on the estimation of volatility functionals. We conduct extensive simulation study to verify the theoretical results. The results demonstrate that our estimators have relative better performance than the existing ones, especially when the jump is of infinite variation. Besides, we apply our estimators to a real high-frequency dataset, which reveals nonzero leverage effect and volatility of volatility in the market.

Paper Structure

This paper contains 24 sections, 11 theorems, 263 equations, 3 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Setup and assumptions
  3. Estimators and asymptotic results
  4. Estimation of spot volatility
  5. Estimation of leverage effect
  6. Estimation of volatility of volatility
  7. Estimation of volatility functional
  8. Feasible central limit theorems and their applications
  9. Discussions
  10. Leverage effect: A correlation perspective
  11. Leverage effect: A volatility functional perspective
  12. The effect of jumps
  13. Simulation studies
  14. Existing leverage effect and volatility of volatility estimators
  15. Simulation design
  16. ...and 9 more sections

Key Result

Theorem 1

Under Assumptions asu-con and asu-cha, and suppose that as $n\rightarrow \infty$, $k_n \rightarrow \infty$, $k_n\Delta_n \rightarrow 0$. Let $k_n = \lfloor \kappa n^{b} \rfloor$ with $0<b<1$ and $\kappa$ a positive constant. (1). If $\max\{\beta,r\}\leq 1$If we select $b=\frac{1}{2}$, then such a co (2). And furthermore, where $U$ is a normal random variable defined on an extension of the origina

Figures (3)

  • Figure 1: The histograms and Q-Q plots for the studentized statistics of leverage effect. In the histograms, the red real line is the density curve of the standard normal distribution, the blue dotted line is the fitted density curve based on the estimates.
  • Figure 2: The histograms and Q-Q plots for the studentized statistics of volatility of volatility. In the histograms, the red real line is the density curve of the standard normal distribution, the blue dotted line is the fitted density curve based on the estimates.
  • Figure 3: The estimates of leverage effect by using monthly data from January 3, 2011, to December 31, 2018, for Apple (APPL), Amazon (AMZN), Intel (INTC), Microsoft (MSFT), and SPDR S$\&$P 500 ETF (SPY).

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 1 more