On the Background Driving Lévy Density Associated with the General Tempered Stable Distribution: Theoretical Properties and Financial Applications

This paper identifies and characterizes the background driving Lévy process (BDLP) associated with the Generalized Tempered Stable (GTS) distribution, a flexible seven-parameter family of infinitely divisible distributions with applications in physics and quantitative finance. We show that the corresponding BDLP is a finite-variation, infinite-activity Type B Lévy process and derive its explicit background driving characteristic exponent function (BDCEF). The resulting BDLP provides a unified representation that encompasses several important special cases, including the bilateral stable, bilateral Gamma, and Variance Gamma distributions. Building on these results, we develop a simulation framework based on a stationary Ornstein–Uhlenbeck (OU)-type process driven by the GTS BDLP. The mean-reversion speed parameter of the OU process is calibrated using maximum likelihood estimation applied to daily return data from the SPY ETF and Ethereum over the period 2010–2024. The proposed simulation methodology produces realistic daily cumulative return trajectories, and comprehensive numerical error analyses demonstrate the accuracy and efficiency of the resulting discretization scheme. These findings provide both a theoretical extension of Lévy-driven OU models and a practical framework for simulating complex financial return dynamics.