Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation

Abstract

We develop 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 of 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 SBG coupling between a Lévy process and its Gaussian approximation. Numerical performance, based on the implementation in the dedicated GitHub repository, exhibits a good agreement with our theoretical bounds.

45 pages, 6 figures, 4 tables, short video on https://youtu.be/EL0v2QUb5tQ

References in corpus (4)

• Ruin Probabilities and Overshoots for General Levy Insurance Risk Processes
• Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction
• Estimation for Lévy processes from high frequency data within a long time interval
• Convex minorants and the fluctuation theory of Lévy processes

Cited by in corpus (2)

• Monte Carlo algorithm for the extrema of tempered stable processes
• Non-Markovianity of $2K-B$ and a degeneration