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A Global Adaptive Quasi-Monte Carlo Algorithm for Functions of Low Truncation Dimension Applied to Problems from Finance

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Book cover Monte Carlo and Quasi-Monte Carlo Methods 2010

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 23))

Abstract

We show how to improve the performance of the quasi-Monte Carlo method for solving some pricing problems from financial engineering. The key point of the new algorithm, coined “GELT”, is an adaptive re-ordering of the point set so that the function is sampled more frequently in the regions where there is greater variation. The adaptivity only operates on the first few dimensions of the integrand and we show how to explicitly obtain the points of a digital sequence falling into boxes into these first few dimensions. This is effective as the problem is first transformed into having “low truncation dimension”. In general it is assumed that finance problems have low effective dimension. In addition we make use of a so-called “sniffer function” to cope with the discontinuity in the integrand function. Numerical results with the new adaptive algorithm are presented for pricing a digital Asian option, an Asian option and an Asian option with an up-and-out barrier.

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Acknowledgements

The authors are grateful to Prof. Ian H. Sloan for useful discussions related to this paper and very much appreciated the careful comments and questions from the two anonymous referees. The first author is a fellow of the Research Foundation Flanders (FWO) and is grateful to the University of New South Wales where large parts of this paper were written; and therefore also thanks the Australian Research Council (ARC) for support.

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Authors and Affiliations

  1. Department of Computer Science, K.U.Leuven, Celestijnenlaan 200A, 3001, Heverlee, Belgium

    Dirk Nuyens

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Dirk Nuyens & Benjamin J. Waterhouse

Authors
  1. Dirk Nuyens
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  2. Benjamin J. Waterhouse
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Corresponding author

Correspondence to Dirk Nuyens .

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Editors and Affiliations

  1. , Institute of Applied Mathematics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland

    Leszek Plaskota

  2. , Institute of Applied Mathematics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland

    Henryk Woźniakowski

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Nuyens, D., Waterhouse, B.J. (2012). A Global Adaptive Quasi-Monte Carlo Algorithm for Functions of Low Truncation Dimension Applied to Problems from Finance. In: Plaskota, L., Woźniakowski, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics & Statistics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27440-4_34

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