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

1. Problem Statement

The paper addresses the statistical inference of two critical quantities in high-frequency financial econometrics: the leverage effect (the covariance between asset returns and volatility) and the volatility of volatility (VoV) (the variation strength of the volatility process).

The core challenge lies in the presence of jumps in the underlying asset price process. Existing estimators for these quantities often rely on thresholding techniques or bi-power variations, which suffer from significant limitations:

• Finite Variation Restriction: Many existing methods are only consistent or asymptotically normal when jumps are of finite variation (Jump Activity Index, JAI $< 1$).
• Infinite Variation Jumps: Real-world high-frequency data often exhibits jumps of infinite variation (JAI $\ge 1$), where standard estimators fail to achieve optimal convergence rates or valid central limit theorems (CLT).
• Bias and Tuning: Thresholding methods require complex parameter tuning dependent on unknown jump activity indices, and they often fail to eliminate the bias caused by "small" jumps effectively.

The authors aim to develop robust estimators that remain valid and efficient even when jumps are of infinite variation (up to JAI $< 2$ for consistency and $< 1.5$ for asymptotic normality).

2. Methodology

The proposed methodology relies on the Empirical Characteristic Function (ECF) of high-frequency increments to separate the continuous volatility component from the jump component.

A. Model Setup

The log-price process $X_t$ is modeled as an Itô semimartingale:
$$X_t = X_0 + \int_0^t b_s ds + \int_0^t \sigma_s dB_s + J_t$$
where $J_t$ is a pure jump process. The jump component is decomposed into a structured Lévy process $L$ (with Blumenthal-Getoor index $\beta$) and a general jump process $J'$ (with activity index $r$). The volatility process $\sigma_t$ is also modeled as a continuous Itô semimartingale.

B. Spot Volatility Estimation

The authors utilize a spot volatility estimator proposed in Liu et al. (2018), constructed via the ECF:
$$\hat{\sigma}^2_t = -\frac{2}{u^2} \log \left( \left[ \frac{1}{k_n} \sum_{j} \cos \left( \frac{u \Delta^n_j X}{\sqrt{\Delta_n}} \right) \right] \vee \frac{1}{\sqrt{k_n}} \right)$$

• Mechanism: The cosine function (part of the characteristic function) dampens the impact of jumps. By taking the logarithm of the average of cosines, the estimator isolates the volatility $\sigma^2_t$.
• Advantage: Unlike thresholding (which removes large jumps but leaves small ones) or bi-power (which reduces jump impact but not to zero), the ECF approach controls the influence of jumps of all sizes, particularly effective for infinite variation jumps.

C. Leverage Effect Estimator

The leverage effect $L_{[0,T]} = \langle X, \sigma^2 \rangle_T$ is estimated by summing the product of price increments and the difference between forward and backward spot volatility estimates:
$$\hat{L}_{[0,T]} = \sum_{i} \Delta^n_i X \cdot (\hat{\sigma}^2_{t_{i+}} - \hat{\sigma}^2_{t_{i-}})$$

• Bias Removal: The difference between forward ($\hat{\sigma}^2_{t_{i+}}$) and backward ($\hat{\sigma}^2_{t_{i-}}$) estimators cancels out the leading-order bias terms induced by jumps, provided the jump structure satisfies specific symmetry or regularity conditions.

D. Volatility of Volatility Estimator

The integrated VoV is defined as $VoV_{[0,T]} = \int_0^T 4\sigma^2_t ((\tilde{\sigma}_t)^2 + (\tilde{\sigma}'_t)^2) dt$. The estimator is constructed using the squared difference of spot volatility estimates, corrected by a term involving the function $h_1$ (derived from the ECF variance):
$$\widehat{VoV}_{[0,T]} = \sum_{i} \left( \frac{3}{2k_n} (\hat{\sigma}^2_{t_{i+}} - \hat{\sigma}^2_{t_{i-}})^2 - \frac{3}{k_n^2} h_1(u, t_{i+}, \hat{\sigma}^2_{t_{i+}}) \right)$$

E. Feasible Central Limit Theorems

To make inference practical, the authors propose consistent estimators for the asymptotic variances of the leverage and VoV estimators. This allows for the construction of feasible central limit theorems, enabling the calculation of confidence intervals and hypothesis testing without knowing the true underlying parameters.

3. Key Contributions

1. Relaxed Jump Conditions: The proposed estimators achieve consistency and asymptotic normality under significantly more general jump conditions than existing literature.
   • Consistency: Valid for JAI $< 2$ (for VoV) and JAI $\le 1$ (for Leverage), whereas previous methods often required JAI $< 1$.
   • Asymptotic Normality: Valid for JAI $< 1.5$ (VoV) and JAI $\le 1$ (Leverage), compared to JAI $< 1/2$ or $< 1/3$ in some existing works.
2. Optimal Convergence Rate: The estimators achieve the optimal convergence rate of $n^{-1/4}$ for both leverage effect and VoV, even in the presence of infinite variation jumps.
3. No Parameter Tuning: Unlike thresholding methods that require tuning parameters ($\alpha, \omega$) dependent on the unknown jump activity index, the proposed method relies on a parameter $u$ which can be fixed (e.g., $u=1$) to yield satisfactory performance without data-driven tuning.
4. Theoretical Refinement: The paper corrects and refines the asymptotic properties of the spot volatility estimator from Liu et al. (2018), proving it achieves the optimal $n^{-1/4}$ rate (rather than the previously stated $n^{-1/4}/\log n$) and providing a feasible CLT (estimating the variance), which was missing in prior work.
5. Generalization: The theory is extended to leverage effects defined as the covariance between price and a general function of volatility $F(\sigma^2)$, showing the proposed estimator has a smaller asymptotic variance than existing continuous-case estimators (e.g., Wang and Mykland, 2014).

4. Results

Theoretical Results

• Proposition 1 & Theorem 1: Established the consistency and asymptotic normality of the leverage effect estimator. The limiting variance is derived explicitly, and a feasible version is constructed.
• Theorem 2: Established consistency and asymptotic normality for the VoV estimator. The paper proves that the bias terms cancel out effectively when taking differences of spot volatility estimates, even for infinite variation jumps.
• Theorem 3-5: Proved the consistency of the volatility functional estimators used to construct the feasible variances.

Simulation Studies

• Performance: Extensive simulations compared the proposed estimators ("Lev-our", "Vov-our") against existing methods (Lev-AJ14, Lev-AFLWY17, Vov-BV09, etc.).
• Infinite Variation Jumps: When jumps were of infinite variation ($\beta = 1.5$), existing estimators showed large biases or failed to converge, while the proposed estimators maintained low bias and Mean Squared Error (MSE).
• Finite Sample: The proposed estimators demonstrated robust performance across various jump activity indices ($\beta = 0.5, 1, 1.5$).
• CLT Verification: Histograms and Q-Q plots confirmed that the studentized statistics converge to the standard normal distribution, validating the feasible CLTs.

Empirical Application

• Data: Applied to 8 years of high-frequency data (2011–2018) for SPY and four constituent stocks (AAPL, AMZN, INTC, MSFT).
• Jump Activity: Estimated Jump Activity Indices (JAI) were found to be frequently $> 0.5$ and often $> 1$ (especially for SPY), confirming the presence of intensive infinite variation jumps in real data.
• Findings:
  • Leverage Effect: Significant negative leverage effects were detected for most assets and years, consistent with financial theory.
  • Volatility of Volatility: Evidence of non-zero VoV was found. The hypothesis test rejected the null hypothesis of zero VoV for most assets in most years, particularly in 2018.
  • Robustness: The method successfully handled the high jump intensity observed in the real data without requiring complex parameter tuning.

5. Significance

This paper provides a robust, theoretically sound, and practically convenient framework for estimating leverage and volatility of volatility in the presence of realistic market microstructure features, specifically infinite variation jumps.

• For Practitioners: It offers a "plug-and-play" solution that does not require the difficult task of estimating jump activity indices to tune parameters, making it superior for empirical studies where jump characteristics are unknown.
• For Researchers: It extends the theoretical boundaries of high-frequency econometrics, proving that optimal inference is possible even when the jump activity is high (infinite variation), a regime where many standard tools break down.
• Market Insight: The empirical application confirms that real-world high-frequency data is dominated by jump activity that invalidates simpler continuous models, and that the proposed estimators are necessary to accurately capture the dynamics of leverage and volatility clustering.