[Paper Review] Diffusion covariation and co-jumps in bidimensional asset price processes with stochastic volatility and infinite activity Levy jumps

Fabio Gobbi, Cecilia Mancini|arXiv (Cornell University)|May 9, 2007

Stochastic processes and financial applications3 citations

TL;DR

This paper proposes a threshold-based estimator to consistently separate diffusion covariation from co-jumps in two-dimensional asset price processes with stochastic volatility and infinite activity Lévy jumps. Using discrete high-frequency data, the method isolates large increments to detect co-jumps and establishes that the convergence rate of the diffusion covariation estimator is √h only under moderate jump activity, not in general infinite activity settings.

ABSTRACT

In this paper we consider two processes driven by diffusions and jumps. The jump components are Levy processes and they can both have finite activity and infinite activity. Given discrete observations we estimate the covariation between the two diffusion parts and the co-jumps. The detection of the co-jumps allows to gain insight in the dependence structure of the jump components and has important applications in finance. Our estimators are based on a threshold principle allowing to isolate the jumps. This work follows Gobbi and Mancini (2006) where the asymptotic normality for the estimator of the covariation, with convergence speed given by the squared root of h, was obtained when the jump components have finite activity. Here we show that the speed is the squared root of h only when the activity of the jump components is moderate.

Motivation & Objective

• To estimate the continuous covariation between two asset price processes driven by diffusions and jumps, separating it from co-jumps.
• To develop a consistent threshold-based estimator that isolates jumps using high-frequency observations.
• To determine the asymptotic convergence rate of the diffusion covariation estimator when jump components have infinite activity.
• To analyze the asymptotic normality of the estimator under stable-like Lévy jump laws and copula dependence structures.
• To extend prior work on finite-activity jumps to the more complex case of infinite-activity Lévy processes with general dependence.

Proposed method

• Uses a threshold criterion to identify intervals where jumps exceed a pre-specified size, based on increment magnitudes of observed price changes.
• Applies a truncated realized quadratic covariation estimator: $\tilde{v}^{(n)}_{1,1}(X^{(1)},X^{(2)})_{T} = \sum_{j=1}^{n} \Delta_j X^{(1)} \mathbf{1}_{\{ (\Delta_j X^{(1)})^2 \leq r(h) \}} \Delta_j X^{(2)} \mathbf{1}_{\{ (\Delta_j X^{(2)})^2 \leq r(h) \}} $, which excludes large increments likely due to jumps.
• Decomposes the jump component into finite-activity jumps $J_1^{(q)}$ and infinite-activity Lévy components $\tilde{J}_2^{(q)}$, assuming stable-like tail behavior.
• Employs a Lindeberg-Feller central limit theorem framework to establish asymptotic normality of the estimator under specific dependence assumptions via copula modeling.
• Derives convergence rates by analyzing the behavior of higher-order moments and tail probabilities of jump increments as sampling frequency increases ($h \to 0$).
• Uses moment bounds and asymptotic expansions to show that the estimator remains consistent even under infinite activity, but the convergence rate $\sqrt{h}$ holds only when jump activity is moderate.

Experimental results

Research questions

• RQ1 What is the convergence rate of the threshold estimator for diffusion covariation when the jump components have infinite activity?
• RQ2 How does the dependence structure of co-jumps—modeled via copulas—impact the asymptotic distribution of the estimator?
• RQ3 Can the threshold-based method consistently separate diffusion covariation from co-jumps in the presence of infinite activity Lévy jumps?
• RQ4 Under what conditions on the jump activity and tail behavior (e.g., stable-like laws) does the estimator achieve asymptotic normality?
• RQ5 How does the inclusion of infinite activity Lévy jumps affect the bias and variance of the realized covariation estimator compared to the finite-activity case?

Key findings

• The threshold estimator $\tilde{v}^{(n)}_{1,1}$ is consistent for the integrated diffusion covariation $\int_0^T \rho_t \sigma_t^{(1)} \sigma_t^{(2)} dt$ under general conditions, including infinite activity Lévy jumps.
• The convergence rate of the estimator is $\sqrt{h}$ only when the jump activity is moderate; for general infinite activity, the rate may be slower depending on the tail indices $\alpha_1, \alpha_2$.
• Asymptotic normality of the estimator is established under the assumption that the infinite activity Lévy components $\tilde{J}_2^{(q)}$ have stable-like laws and the joint dependence is governed by a copula in a given class.
• The Lindeberg condition is verified by showing that the probability of large deviations in the jump increment terms vanishes as $h \to 0$, ensuring convergence to normality.
• The method successfully isolates co-jumps by excluding intervals with large increments, reducing bias in the covariation estimate compared to standard realized covariation.
• Theoretical results show that the contribution of the infinite activity jump components to the estimation error diminishes at a rate dependent on the stability indices $\alpha_1, \alpha_2$ and the sampling frequency $h$.