Summary: TESTING WHETHER JUMPS HAVE
FINITE OR INFINITE ACTIVITY
By Yacine A´t-Sahalia1 and Jean Jacod
Princeton University and UPMC (UniversitÚ Paris-6)
We propose statistical tests to discriminate between the finite and infinite
activity of jumps in a semimartingale discretely observed at high frequency.
The two statistics allow for a symmetric treatment of the problem: we can
either take the null hypothesis to be finite activity, or infinite activity. When
implemented on high frequency stock returns, both tests point towards the
presence of infinite activity jumps in the data.
1. Introduction. Traditionally, models with jumps in finance have relied on
Poisson processes, as in Merton (1976), Ball and Torous (1983) and Bates (1991).
These jump-diffusion models allow for a finite number of jumps in a finite time
interval, with the idea that the Brownian-driven diffusive part of the model cap-
tures normal asset price variations while the Poisson-driven jump part of the model
captures large market moves in response to unexpected information. More recently,
financial models have been proposed, that allow for infinitely many jumps in finite
time intervals, using a variety of specifications, such as the variance gamma model of
Madan and Seneta (1990) and Madan et al. (1998), the hyperbolic model of Eberlein
and Keller (1995), the CGMY model of Carr et al. (2002) and the finite moment log