Monte Carlo algorithm for the extrema of tempered stable processes

TL;DR

This paper introduces a fast Monte Carlo algorithm for accurately estimating the extrema of tempered stable Lévy processes, with applications in option pricing and risk measurement, supported by theoretical proofs and numerical tests.

Contribution

The paper presents a new Monte Carlo method for extrema of tempered stable processes with proven geometric convergence and optimal complexity, including a CLT for confidence interval construction.

Findings

Algorithm converges geometrically fast for discontinuous functions

Multilevel Monte Carlo estimator achieves optimal $ ext{O}( ext{ε}^{-2})$ complexity

Numerical examples demonstrate superior performance over existing methods

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 L\'evy 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 guidelines for users to the best method for a given set of parameters. We illustrate the performance of the algorithm with numerical examples.