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Evaluating Expectations of Functionals of Brownian Motions: a Multilevel Idea

Authors: Fred J. Hickernell, Thomas Müller-Gronbach, Ben Niu, and Klaus Ritter

Published in: Dagstuhl Seminar Proceedings, Volume 9391, Algorithms and Complexity for Continuous Problems (2009)

Abstract

Prices of path dependent options may be modeled as expectations of functions of an infinite sequence of real variables. This talk presents recent work on bounding the error of such expectations using quasi-Monte Carlo algorithms. The expectation is approximated by an average of $n$ samples, and the functional of an infinite number of variables is approximated by a function of only $d$ variables. A multilevel algorithm employing a sum of sample averages, each with different truncated dimensions, $d_l$, and different sample sizes, $n_l$, yields faster convergence than a single level algorithm. This talk presents results in the worst-case error setting.

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Fred J. Hickernell, Thomas Müller-Gronbach, Ben Niu, and Klaus Ritter. Evaluating Expectations of Functionals of Brownian Motions: a Multilevel Idea. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 9391, pp. 1-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{hickernell_et_al:DagSemProc.09391.3,
  author =	{Hickernell, Fred J. and M\"{u}ller-Gronbach, Thomas and Niu, Ben and Ritter, Klaus},
  title =	{{Evaluating Expectations of Functionals of Brownian Motions: a Multilevel Idea}},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  pages =	{1--19},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{9391},
  editor =	{Thomas M\"{u}ller-Gronbach and Leszek Plaskota and Joseph. F. Traub},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09391.3},
  URN =		{urn:nbn:de:0030-drops-22987},
  doi =		{10.4230/DagSemProc.09391.3},
  annote =	{Keywords: Brownian motions, multilevel, option pricing, worst-case error}
}

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Evaluating Expectations of Functionals of Brownian Motions: a Multilevel Idea

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