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Multilevel Monte Carlo for Asian options and limit theorems

Author

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  • Ben Alaya Mohamed
  • Kebaier Ahmed

    (Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 99, av. J.B. Clément, 93430 Villetaneuse, France)

Abstract

The purpose of this paper is to study the problem of pricing Asian options using the multilevel Monte Carlo method recently introduced by Giles [Oper. Res. 56 (2008), no. 3, 607–617] and to prove a central limit theorem of Lindeberg–Feller type for the obtained algorithm. Indeed, the implementation of such a method requires first a discretization of the integral of the payoff process. For this, we use two well-known second order discretization schemes, namely, the Riemann scheme and the trapezoidal scheme. More precisely, for each of these schemes, we prove a stable law convergence result for the error on two consecutive levels of the algorithm. This allows us to go further and prove two central limit theorems on the multilevel algorithm providing us a precise description on the choice of the associated parameters with an explicit representation of the limiting variance. For this setting of second order schemes, we give new optimal parameters leading to the convergence of the central limit theorem. The complexity of the multilevel Monte Carlo algorithm will be determined.

Suggested Citation

  • Ben Alaya Mohamed & Kebaier Ahmed, 2014. "Multilevel Monte Carlo for Asian options and limit theorems," Monte Carlo Methods and Applications, De Gruyter, vol. 20(3), pages 181-194, September.
    • Handle: RePEc:bpj:mcmeap:v:20:y:2014:i:3:p:181-194:n:2
    DOI: 10.1515/mcma-2013-0025
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    References listed on IDEAS

    as
    1. Michael Giles & Desmond Higham & Xuerong Mao, 2009. "Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff," Finance and Stochastics, Springer, vol. 13(3), pages 403-413, September.
    2. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    3. Bernard Lapeyre & Emmanuel Temam, 2001. "Competitive Monte Carlo methods for the pricing of Asian options," Post-Print hal-01667057, HAL.
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    Cited by:

    1. Desmond J. Higham, 2015. "An Introduction to Multilevel Monte Carlo for Option Valuation," Papers 1505.00965, arXiv.org.

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