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

Abstract

In this paper we consider two processes driven by diffusions and jumps. The jump components are Lévy 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 √ h, was obtained when the jump components have finite activity. Here we show that the speed is √ h only when the activity of the jump components is moderate.

Non‐parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps

“…Cont & Mancini (2008) use TC to test for the presence of a diffusion component in asset prices. Gobbi & Mancini (2006, 2007) use TC to disentangle the co‐jumps and the correlation between the diffusion parts of two semimartingales with Lévy type jumps. Aït‐Sahalia & Jacod use TC to estimate the Blumenthal–Getoor index of J (Aït‐Sahalia & Jacod, 2007) and to estimate the local volatility values and to reach a test for the presence of jumps in a discretely observed semimartingale (Aït‐Sahalia & Jacod, 2008).…”

Modelling systemic price cojumps with Hawkes factor models

“…However, there has been less research on cojumps (i.e. simultaneous jumps in two or more stock prices) and most of them are of empirical nature such as [21,22,23,24]; exceptions are the papers of [25,26] and [27] on cojump estimation and modelling.…”

Truncated Realized Covariance when prices have infinite variation jumps

“…Remarks on the last result. i) Condition α 2 < α 1 (1/u − 1) is equivalent to u < α 1 /(α 2 + α 1 ) and we did not include it among the ones in (7) because such conditions are required for the convergence of some terms of I 4 (defined within the proof of the Theorem) inÎC − IC, while α 1 /(α 2 + α 1 ) is only a separator to establish whether the leading term is ε 1+α 2 /α 1 −α 2 or hε −α 2 . There is another proof for the convergence of some of the cited terms of I 4 , which avoids conditions (7), but it is much longer than the one given in Appendix 1.…”

“…Since it is the same for γ ∈ (0, 1) or γ = 0, let us take γ = 0. For fixed h and u, define R the region identified by the initial assumptions on α 1 , α 2 and by (7) and A, B, C the subregions identified respectively in (10), (11) and (12):…”