Modul:   MAT959  Seminar in Data Science and Mathematical Modelling

Geometrically Convergent Simulation of the Extrema of Lévy Processes

Vortrag von Prof. Dr. Aleksandar Mijatović

Sprecher eingeladen von: Prof. Dr. Delia Marina Coculescu

Datum: 12.10.23  Zeit: 12.15 - 13.45  Raum: Y27H12

Abstract: In this talk, I will describe a computational method for expected functionals of the drawdown and its duration in exponential Lévy models. It is based on a novel simulation algorithm for the joint law of the state, supremum and time the supremum is attained of the Gaussian approximation for a general Lévy process. We bound the bias for various locally Lipschitz and discontinuous payoffs arising in applications and analyse the computational complexities of the corresponding Monte Carlo and multilevel Monte Carlo estimators. Monte Carlo methods for Lévy processes (using Gaussian approximation) have been analysed for Lipschitz payoffs, in which case the computational complexity of our algorithm is up to two orders of magnitude smaller when the jump activity is high. At the core of our approach are bounds on certain Wasserstein distances, obtained via the novel stick-breaking Gaussian (SBG) coupling between a Lévy process and its Gaussian approximation. Numerical performance (based on the implementation in the GitHub repository https://github.com/jorgeignaciogc/SBG.jl) exhibits a good agreement with our theoretical bounds. Numerical evidence suggests that our algorithm remains stable and accurate when estimating Greeks for barrier options and outperforms the “obvious” algorithm for finite-jump-activity Lévy processes. This is joint work with J. Gonzalez-Cázares.