Multilevel Monte Carlo simulation for Lévy processes based on the Wiener–Hopf factorisation

Ferreiro-Castilla, Albert; Kyprianou, Andreas E.; Scheichl, Robert; Suryanarayana, Gowri

doi:10.1016/j.spa.2013.09.015

Cited by 34 publications

(58 citation statements)

References 31 publications

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“…It is inferred from [13] that 142 A. FERREIRO-CASTILLA AND K. VAN SCHAIK any functional applied to the random walk generated in (7) satisfies condition (iii) of Theorem 4. The Lipschitz assumption on f and the triangle inequality together with Theorem 3 ensure that β = 1 and, therefore, up to logarithms, the multilevel Monte Carlo estimator of f (τ u ∧ t) provided by Theorem 4 is optimal.…”

Section: Multilevel Monte Carlo Schemes For the First Passage Timementioning

confidence: 98%

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Applying the Wiener-Hopf Monte Carlo Simulation Technique for Lévy Processes to Path Functionals

Ferreiro-Castilla¹,

Schaik²

2015

J. Appl. Probab.

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In this paper we apply the recently established Wiener-Hopf Monte Carlo simulation technique for Lévy processes from Kuznetsov et al. (2011) to path functionals; in particular, first passage times, overshoots, undershoots, and the last maximum before the passage time. Such functionals have many applications, for instance, in finance (the pricing of exotic options in a Lévy model) and insurance (ruin time, debt at ruin, and related quantities for a Lévy insurance risk process). The technique works for any Lévy process whose running infimum and supremum evaluated at an independent exponential time can be sampled from. This includes classic examples such as stable processes, subclasses of spectrally one-sided Lévy processes, and large new families such as meromorphic Lévy processes. Finally, we present some examples. A particular aspect that is illustrated is that the Wiener-Hopf Monte Carlo simulation technique (provided that it applies) performs much better at approximating first passage times than a 'plain' Monte Carlo simulation technique based on sampling increments of the Lévy process.

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Section: Multilevel Monte Carlo Schemes For the First Passage Timementioning

confidence: 98%

“…In [13] a comprehensive error analysis is carried out for the bivariate distribution (X g(n,n/t) ,X g(n,n/t) ) from where convergence rates are derived in terms of the moments of g(n, n/t). For a Lévy process with finite second moment it is deduced that E[(X g(n,n/t) − X t ) 2 ] = O(n −1/2 ).…”

Section: Remarkmentioning

confidence: 99%

Applying the Wiener-Hopf Monte Carlo Simulation Technique for Lévy Processes to Path Functionals

Ferreiro-Castilla¹,

Schaik²

2015

J. Appl. Probab.

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“…He introduced a more efficient Multi Level quasi-Monte Carlo method [3]. Recently in 2014, Ferreiro-Castilla et al presented an article on multilevel Monte Carlo simulation for Levy processes based on Wiener-Hopf factorization [10].…”

Section: Theoretical Backgrounds and An Overview Of The Research Historymentioning

confidence: 99%

The pricing of spread option using simulation

Erfanian

Bidgoli

Shakibaei³

2017

IJAMR

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Monte Carlo simulation is one of the most common and popular method of options pricing. The advantages of this method are being easy to use, suitable for all kinds of standard and exotic options and also are suitable for higher dimensional problems. But on the other hand Monte Carlo variance convergence rate is O(n −1 2 ⁄ )which due to that it will have relatively slow convergence rate to answer the problems, as to achieve ε accuracy when it has been d-dimensions, complexity is O(dε −3 ). For this purpose, several methods are provided in quasi Monte Carlo simulation to increase variance convergence rate as variance reduction techniques, so far. One of the latest presented methods is multilevel Monte Carlo that is introduced by Giles in 2008. This method not only reduces the complexity of computing amount O(ε −3 log ( 2 )) in use of Euler discretization scheme and the amount O(ε −2 ) in use of Milstein discretization scheme, but also has the ability to combine with other variance reduction techniques. In this paper, using Multilevel Monte Carlo method by taking Milstein discretization scheme, pricing spread option and compared complexity of computing with standard Monte Carlo method. The results of Multilevel Monte Carlo method in pricing spread options are better than standard Monte Carlo simulation.

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“…Our method involves repeated integrations with respect to the resolvent measure, which is written in terms of the scale function of the underlying spectrally negative Lévy process. For the randomization methods applied in the pricing of finite-time horizon American options, we refer the reader to [28,34]; similar ideas are also used in recent work on the so-called Wiener-Hopf simulation [31,23]. Bouchard et al [10] analyze a maturity randomization algorithm and apply it to stochastic control problems with applications to optimal single stopping and dynamic hedging under uncertain volatility.…”

Section: Introductionmentioning

confidence: 99%

An Analytic Recursive Method for Optimal Multiple Stopping: Canadization and Phase-Type Fitting

Leung

Yamazaki

Zhang

2015

Int. J. Theor. Appl. Finan.

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ABSTRACT. We study an optimal multiple stopping problem for call-type payoff driven by a spectrally negative Lévy process. The stopping times are separated by constant refraction times, and the discount rate can be positive or negative. The computation involves a distribution of the Lévy process at a constant horizon and hence the solutions in general cannot be attained analytically. Motivated by the maturity randomization (Canadization) technique by Carr [14], we approximate the refraction times by independent, identically distributed Erlang random variables. In addition, fitting random jumps to phase-type distributions, our method involves repeated integrations with respect to the resolvent measure written in terms of the scale function of the underlying Lévy process. We derive a recursive algorithm to compute the value function in closed form, and sequentially determine the optimal exercise thresholds. A series of numerical examples are provided to compare our analytic formula to results from Monte Carlo simulation.

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Multilevel Monte Carlo simulation for Lévy processes based on the Wiener–Hopf factorisation

Cited by 34 publications

References 31 publications

Applying the Wiener-Hopf Monte Carlo Simulation Technique for Lévy Processes to Path Functionals

Applying the Wiener-Hopf Monte Carlo Simulation Technique for Lévy Processes to Path Functionals

The pricing of spread option using simulation

An Analytic Recursive Method for Optimal Multiple Stopping: Canadization and Phase-Type Fitting

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