First exit-time analysis for an approximate Barndorff-Nielsen and Shephard model with stationary self-decomposable variance process

TL;DR

This paper analyzes the first exit time in an approximate Barndorff-Nielsen and Shephard model with stationary self-decomposable variance, deriving explicit density functions and applying results to S&P 500 data.

Contribution

It introduces an approximate model with explicit first exit time density functions for specific Le9vy subordinators, linking theory to empirical data.

Findings

Derived explicit probability density functions for first exit times.

Decomposed first exit time into Brownian and Le9vy subordinator components.

Validated model with empirical S&P 500 dataset.

Abstract

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.