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Quasi-Monte Carlo methods with applications in finance

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  • Published: 11 July 2009
  • Volume 13, pages 307–349, (2009)
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Quasi-Monte Carlo methods with applications in finance
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  • Pierre L’Ecuyer1 

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Abstract

We review the basic principles of quasi-Monte Carlo (QMC) methods, the randomizations that turn them into variance-reduction techniques, the integration error and variance bounds obtained in terms of QMC point set discrepancy and variation of the integrand, and the main classes of point set constructions: lattice rules, digital nets, and permutations in different bases. QMC methods are designed to estimate s-dimensional integrals, for moderate or large (perhaps infinite) values of s. In principle, any stochastic simulation whose purpose is to estimate an integral fits this framework, but the methods work better for certain types of integrals than others (e.g., if the integrand can be well approximated by a sum of low-dimensional smooth functions). Such QMC-friendly integrals are encountered frequently in computational finance and risk analysis. We summarize the theory, give examples, and provide computational results that illustrate the efficiency improvement achieved. This article is targeted mainly for those who already know Monte Carlo methods and their application in finance, and want an update of the state of the art on quasi-Monte Carlo methods.

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  1. DIRO, Université de Montréal, C.P. 6128, succ. Centre-Ville, Montréal, QC, H3C 3J7, Canada

    Pierre L’Ecuyer

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L’Ecuyer, P. Quasi-Monte Carlo methods with applications in finance. Finance Stoch 13, 307–349 (2009). https://doi.org/10.1007/s00780-009-0095-y

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  • Received: 30 May 2008

  • Accepted: 20 August 2008

  • Published: 11 July 2009

  • Issue Date: September 2009

  • DOI: https://doi.org/10.1007/s00780-009-0095-y

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Keywords

  • Monte Carlo
  • Quasi-Monte Carlo
  • Variance reduction
  • Effective dimension
  • Discrepancy
  • Hilbert spaces

Mathematics Subject Classification (2000)

  • 65C05
  • 68U20
  • 91B28

JEL Classification

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