  In this paper, an approximate version of the Barndorff-Nielsen and Shephard
model, driven by a Brownian motion and a L\'evy subordinator, is formulated.
The first-exit time of the log-return process for this model is analyzed. It is
shown that with certain probability, the first-exit time process of the
log-return is decomposable into the sum of the first exit time of the Brownian
motion with drift, and the first exit time of a L\'evy subordinator with drift.
Subsequently, the probability density functions of the first exit time of some
specific L\'evy subordinators, connected to stationary, self-decomposable
variance processes, are studied. Analytical expressions of the probability
density function of the first-exit time of three such L\'evy subordinators are
obtained in terms of various special functions. The results are implemented to
empirical S&P 500 dataset.
