This repository was developed for the final project of the course of Financial Engineering at Politecnico di Milano
The aim of this project is to explore the simulation of two broad categories of mean-reverting stochastic processes: Ornstein-Uhlenbeck (OU-Lévy) and Lévy-Ornstein-Uhlenbeck (Lévy-OU). Specifically, we focus on simulating Ornstein-Uhlenbeck Tempered Stable (OU-TS) and Tempered Stable Ornstein- Uhlenbeck (TS-OU) processes. To achieve this, we employ both the Exact Decomposition algorithm based on the property of self-decomposability(asdone in Sabino[4] and SabinoCufaro[5]) and the Fast General Monte Carlo method which requires only the characteristic function of the process (Baviera & Manzoni [2]). The ultimate goal of our work is to utilize these simulations to price energy European and American call options, since these kind of processes are able to capture the dynamics of energy markets.
We have developed a library in MATLAB and in Python in order to meet the requirements for the processes under study. For a better comprehension, the main script does not include all the tests conducted, but it does fulfill the assignment’s requirements. We focused our attention on the optimization in the MATLAB code. The results reported in the PDF document Simulation of Lévy-driven OU Processes Report are obtained using MATLAB; in Python the results are available in the Jupyter notebook.
[1] Baviera, R. & Manzoni, P. (2024). Fast and General Simulation of Lévy-driven OU processes for Energy Derivatives. Preprint arXiv: :2401.15483.
[2] Longstaff, F. A., & Schwartz, E. S. (2001). Valuing American options by simulation: a simple least-squares approach. The review of financial studies, 14(1), 113-147.
[3] Sabino, P. (2022). Pricing energy derivatives in markets driven by tempered stable and CGMY processes of Ornstein–Uhlenbeck type. Risks, 10(8):148.
[4] Sabino, P. & Cufaro Petroni, N. (2022). Fast simulation of tempered stable Ornstein–Uhlenbeck processes. Computational Statistics, 37(5):2517–2551.