Exact simulation of normal tempered stable processes of OU type with applications

Abstract

We study the Ornstein-Uhlenbeck process having a symmetric normal tempered stable stationary law and represent its transition distribution in terms of the sum of independent laws. In addition, we write the background driving Lévy process as the sum of two independent Lévy components. Accordingly, we can design two alternate algorithms for the simulation of the skeleton of the Ornstein-Uhlenbeck process. The solution based on the transition law turns out to be faster since it is based on a lower number of computational steps, as confirmed by extensive numerical experiments. We also calculate the characteristic function of the transition density which is instrumental for the application of the FFT-based method of Carr and Madan (J Comput Finance 2:61–73, 1999) to the pricing of a strip of call options written on markets whose price evolution is modeled by such an Ornstein-Uhlenbeck dynamics. This setting is indeed common for spot prices in the energy field. Finally, we show how to extend the range of applications to future markets.

Point process simulation of generalised hyperbolic Lévy processes

“…Other relevant developments include [48] which present the theory of NIG processes, [49] which present approximate sampling methods for the NIG case, and [50] which give applications of shot -noise series based methods for non-Gaussian Ornstein-Uhlenbeck (OU) processes. Exact simulation methods for the class of tempered stable (TS) processes are studied in ( [51]- [54]). In addition, approximate simulation methods for GH Lévy fields, which are infinite-dimensional GH Lévy processes, are studied in [55].…”

Point process simulation of generalised hyperbolic Lévy processes

“…Other relevant developments include (Barndorff-Nielsen 1997a) which present the theory of NIG processes, Rydberg (1997) which present approximate sampling methods for the NIG case, and Barndorff-Nielsen and Shephard (2001) which give applications of shot-noise series based methods for non-Gaussian Ornstein-Uhlenbeck (OU) processes. Exact simulation methods for the class of tempered stable (TS) processes are studied in Zhang (2011), Qu et al (2021), Grabchak (2019), Sabino (2022). In addition, approximate simulation methods for GH Lévy fields, which are infinite-dimensional GH Lévy processes, are studied in Barth and Stein (2017).…”