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

Jorge González Cázares () and Aleksandar Mijatović ()
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Jorge González Cázares: University of Warwick and The Alan Turing Institute
Aleksandar Mijatović: University of Warwick and The Alan Turing Institute

Finance and Stochastics, 2022, vol. 26, issue 4, No 1, 732 pages

Abstract: 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 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 Cázares and Mijatović (SBG approximation. GitHub repository. Available online at https://github.com/jorgeignaciogc/SBG.jl (2020)), 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.

Keywords: Lévy models; Drawdown; Duration of drawdown; Barrier options; Gaussian approximation; Wasserstein and Kolmogorov bounds for maximum and the time maximum is attained; Simulation; Multilevel Monte Carlo; 60G51; 60G70; 65C05; 91G60 (search for similar items in EconPapers)
JEL-codes: C63 G12 G17 G32 (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s00780-022-00486-7

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