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A Monte Carlo algorithm for the extrema of tempered stable processes

Part of: Stochastic processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  30 June 2023

Jorge Ignacio González Cázares Open the ORCID record for Jorge Ignacio González Cázares [Opens in a new window]  and
Aleksandar Mijatović Open the ORCID record for Aleksandar Mijatović [Opens in a new window]
Jorge Ignacio González Cázares*
Affiliation:
University of Warwick and The Alan Turing Institute
Aleksandar Mijatović*
Affiliation:
University of Warwick and The Alan Turing Institute
*
*Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, United Kingdom.
*Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, United Kingdom.
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Abstract

We develop a novel Monte Carlo algorithm for the vector consisting of the supremum, the time at which the supremum is attained, and the position at a given (constant) time of an exponentially tempered Lévy process. The algorithm, based on the increments of the process without tempering, converges geometrically fast (as a function of the computational cost) for discontinuous and locally Lipschitz functions of the vector. We prove that the corresponding multilevel Monte Carlo estimator has optimal computational complexity (i.e. of order $\varepsilon^{-2}$ if the mean squared error is at most $\varepsilon^2$) and provide its central limit theorem (CLT). Using the CLT we construct confidence intervals for barrier option prices and various risk measures based on drawdown under the tempered stable (CGMY) model calibrated/estimated on real-world data. We provide non-asymptotic and asymptotic comparisons of our algorithm with existing approximations, leading to rule-of-thumb principles guiding users to the best method for a given set of parameters. We illustrate the performance of the algorithm with numerical examples.

Keywords

SupremumtemperedLévy processsimulationMonte Carlo estimationgeometric convergencemultilevel Monte Carlo

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 65C05: Monte Carlo methods
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 4 , December 2023 , pp. 1362 - 1389
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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