Multilevel Monte Carlo for exponential Lévy models

Giles, Michael B.; Xia, Yuan

doi:10.1007/s00780-017-0341-7

Cited by 19 publications

(18 citation statements)

References 27 publications

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“…In the case of no jumps (X is a Brownian motion with drift) boundedness of exponential moments was established in [3]. Furthermore, if X is a b.v. spectrallynegative (-positive) process, then the error M − M (n) is bounded from above by |γ |n −1 , showing that E(M − M (n) ) p = O(n −p ), see also [33]. be the analogue of V (n) but for a shifted grid (i + s)/n with i ∈ Z and all points in [0, 1], we note that also {E V…”

Section: Comments and Extensionsmentioning

confidence: 90%

“…Importantly, the result [21] cannot be generalized in a straightforward fashion to p = 1, since it crucially relies on Spitzer's identity. Nevertheless, [33] provides some bounds for p = 1 in the particular case when σ = 0, γ = 0 and Π(dx) has a density sandwiched between c 1 |x| −1−α and c 2 |x| −1−α for small |x|. These bounds, however, have suboptimal rates (in the logarithmic sense) unless p > 2α or X is spectrally negative.…”

Section: Moments Of the Discretization Errormentioning

confidence: 99%

“…as n → ∞ for any fixed x > 0, which is the probability of failure in detecting threshold exceedance when restricting to the grid time-points. On the way towards this goal, we also provide asymptotics of the moments E(M − M (n) ) p of the discretization error in the approximation of the supremum, markedly improving on the bounds in [33] and other works.…”

Section: Introductionmentioning

confidence: 99%

“…This sparkled research in various application areas including mathematical finance, see [15] for approximations of option prices in discrete-time models using continuoustime counterparts. Various expansions and bounds on the expected error E(M − M (n) ) were derived in [21,33,35] among others. The error probability ∆ n (x) asymptotics was identified in [14] in the case of a linear Brownian motion and later extended in [23] to Brownian motion perturbed by an independent compound Poisson process.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Zooming-in on a Lévy process: failure to observe threshold exceedance over a dense grid

Bisewski¹,

Ivanovs²

2020

Electron. J. Probab.

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For a Lévy process X on a finite time interval consider the probability that it exceeds some fixed threshold x > 0 while staying below x at the points of a regular grid. We establish exact asymptotic behavior of this probability as the number of grid points tends to infinity. We assume that X has a zooming-in limit, which necessarily is 1/α-self-similar Lévy process with α ∈ (0, 2], and restrict to α > 1. Moreover, the moments of the difference of the supremum and the maximum over the grid points are analyzed and their asymptotic behavior is derived. It is also shown that the zooming-in assumption implies certain regularity properties of the ladder process, and the decay rate of the left tail of the supremum distribution is determined.

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Section: Comments and Extensionsmentioning

confidence: 90%

Section: Moments Of the Discretization Errormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Zooming-in on a Lévy process: failure to observe threshold exceedance over a dense grid

Bisewski¹,

Ivanovs²

2020

Electron. J. Probab.

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“…The degree of coupling is usually measured in terms of the variance Var[ f (X l T ) − f (X l−1 T )]. It is shown in Giles [7] (see also Giles and Xia [8]), that under the assumptions…”

Section: Introductionmentioning

confidence: 99%

Multilevel path simulation for weak approximation schemes with application to Lévy-driven SDEs

Belomestny¹,

Nagapetyan²

2017

Bernoulli

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In this paper we discuss the possibility of using multilevel Monte Carlo (MLMC) methods for weak approximation schemes. It turns out that by means of a simple coupling between consecutive time discretisation levels, one can achieve the same complexity gain as under the presence of a strong convergence. We exemplify this general idea in the case of weak Euler scheme for Lévy driven stochastic differential equations, and show that, given a weak convergence of order α ≥ 1/2, the complexity of the corresponding "weak" MLMC estimate is of order −2 log 2 ( ). The numerical performance of the new "weak" MLMC method is illustrated by several numerical examples.

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

Cázares

Mijatović

2022

Finance Stoch

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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.

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Multilevel Monte Carlo for exponential Lévy models

Cited by 19 publications

References 27 publications

Zooming-in on a Lévy process: failure to observe threshold exceedance over a dense grid

Zooming-in on a Lévy process: failure to observe threshold exceedance over a dense grid

Multilevel path simulation for weak approximation schemes with application to Lévy-driven SDEs

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

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