Truncated Realized Covariance when prices have infinite variation jumps

The speed of convergence of the truncated realized covariance to the integrated covariation between the two Brownian parts of two semimartingales is heavily influenced by the presence of infinite activity jumps with infinite variation. Namely, the two processes small jumps play a crucial role through their degree of dependence, other than through their jump activity indices. This theoretical result is established when the semimartingales are observed discretely on a finite time horizon. The estimator in many cases is less efficient than when the model only has finite variation jumps. The small jumps of each semimartingale are assumed to be the small jumps of a Lévy stable process, and to the two stable processes a parametric simple dependence structure is imposed, which allows to range from independence to monotonic dependence. The result of this paper is relevant in financial economics, since by the truncated realized covariance it is possible to separately estimate the common jumps among assets, which has important implications in risk management and contagion modeling.