{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T14:53:08Z","timestamp":1781016788610,"version":"3.54.1"},"reference-count":15,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,6,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Many pricing problems boil down to the computation of a high-dimensional integral, which is usually estimated using Monte Carlo. In fact, the accuracy of a Monte Carlo estimator with <jats:italic>M<\/jats:italic> simulations is given by <jats:inline-formula id=\"j_mcma-2020-2081_ineq_9999_w2aab3b7d174b1b6b1aab1c14b1b3Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mfrac>\n                              <m:mi>\u03c3<\/m:mi>\n                              <m:msqrt>\n                                 <m:mi>M<\/m:mi>\n                              <\/m:msqrt>\n                           <\/m:mfrac>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_mcma-2020-2081_eq_0111.png\"\/>\n                        <jats:tex-math>{\\frac{\\sigma}{\\sqrt{M}}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. Meaning that its convergence is immune to the dimension of the problem. However, this convergence can be relatively slow depending on the variance \u03c3 of the function to be integrated. To resolve such a problem, one would perform some variance reduction techniques such as importance sampling, stratification, or control variates. In this paper, we will study two approaches for improving the convergence of Monte Carlo using Neural Networks. The first approach relies on the fact that many high-dimensional financial problems are of low effective dimensions.\nWe expose a method to reduce the dimension of such problems in order to keep only the necessary variables. The integration can then be done using fast numerical integration techniques such as Gaussian quadrature. The second approach consists in building an automatic control variate using neural networks. We learn the function to be integrated (which incorporates the diffusion model plus the payoff function) in order to build a network that is highly correlated to it. As the network that we use can be integrated exactly, we can use it as a control variate.<\/jats:p>","DOI":"10.1515\/mcma-2020-2081","type":"journal-article","created":{"date-parts":[[2021,1,23]],"date-time":"2021-01-23T03:03:35Z","timestamp":1611371015000},"page":"91-104","source":"Crossref","is-referenced-by-count":4,"title":["Automatic control variates for option pricing using neural networks"],"prefix":"10.1515","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9662-0452","authenticated-orcid":false,"given":"Zineb","family":"El Filali Ech-Chafiq","sequence":"first","affiliation":[{"name":"Universit\u00e9 Grenoble Alpes , CNRS , Grenoble INP, LJK, 38000 Grenoble ; and Quantitative Analyst at Natixis, Paris , France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J\u00e9r\u00f4me","family":"Lelong","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Grenoble Alpes , CNRS , Grenoble INP, LJK, 38000 Grenoble , France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Adil","family":"Reghai","sequence":"additional","affiliation":[{"name":"Head of Quantitative Research, Equity and Commodity Markets , Natixis , 47 quai d\u2019Austerlitz, 75013 Paris , France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2021,1,13]]},"reference":[{"key":"2021053122092006445_j_mcma-2020-2081_ref_001_w2aab3b7d174b1b6b1ab2ab1Aa","unstructured":"M.  Cilimkovic,\nNeural networks and back propagation algorithm,\npreprint (2010), http:\/\/www.dataminingmasters.com\/uploads\/studentProjects\/NeuralNetworks.pdf."},{"key":"2021053122092006445_j_mcma-2020-2081_ref_002_w2aab3b7d174b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"C.  De Luigi, J.  Lelong and S.  Maire,\nRobust adaptive numerical integration of irregular functions with applications to basket and other multi-dimensional exotic options,\nAppl. Numer. Math. 100 (2016), 14\u201330.","DOI":"10.1016\/j.apnum.2015.11.001"},{"key":"2021053122092006445_j_mcma-2020-2081_ref_003_w2aab3b7d174b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"P. W.  Glynn and R.  Szechtman,\nSome new perspectives on the method of control variates,\nMonte Carlo and Quasi-Monte Carlo Methods (Hong Kong 2000),\nSpringer, Berlin (2002), 27\u201349.","DOI":"10.1007\/978-3-642-56046-0_3"},{"key":"2021053122092006445_j_mcma-2020-2081_ref_004_w2aab3b7d174b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"A.  Kebaier and J.  Lelong,\nCoupling importance sampling and multilevel Monte Carlo using sample average approximation,\nMethodol. Comput. Appl. Probab. 20 (2018), no. 2, 611\u2013641.","DOI":"10.1007\/s11009-017-9579-y"},{"key":"2021053122092006445_j_mcma-2020-2081_ref_005_w2aab3b7d174b1b6b1ab2ab5Aa","unstructured":"S.  Kim and S. G.  Henderson,\nAdaptive control variates,\nProceedings of the 2004 Winter Simulation Conference,\nIEEE Press, Piscataway (2004), 621\u2013629."},{"key":"2021053122092006445_j_mcma-2020-2081_ref_006_w2aab3b7d174b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"S.  Kim and S. G.  Henderson,\nAdaptive control variates for finite-horizon simulation,\nMath. Oper. Res. 32 (2007), no. 3, 508\u2013527.","DOI":"10.1287\/moor.1070.0251"},{"key":"2021053122092006445_j_mcma-2020-2081_ref_007_w2aab3b7d174b1b6b1ab2ab7Aa","unstructured":"D. P.  Kingma and J. L.  Ba,\nAdam: A method for stochastic optimization,\npreprint (2017), https:\/\/arxiv.org\/abs\/1412.6980;\nICLR, 2015."},{"key":"2021053122092006445_j_mcma-2020-2081_ref_008_w2aab3b7d174b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"V.  K\u00c5\u00afrkov\u00e1,\nKolmogorov\u2019s theorem and multilayer neural networks,\nNeural Networks 5 (1992), no. 3, 501\u2013506.","DOI":"10.1016\/0893-6080(92)90012-8"},{"key":"2021053122092006445_j_mcma-2020-2081_ref_009_w2aab3b7d174b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"B.  Lapeyre and J.  Lelong,\nA framework for adaptive Monte Carlo procedures,\nMonte Carlo Methods Appl. 17 (2011), no. 1, 77\u201398.","DOI":"10.1515\/mcma.2011.002"},{"key":"2021053122092006445_j_mcma-2020-2081_ref_010_w2aab3b7d174b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"S.  Maire,\nReducing variance using iterated control variates,\nJ. Stat. Comput. Simul. 73 (2003), no. 1, 1\u201329.","DOI":"10.1080\/00949650215726"},{"key":"2021053122092006445_j_mcma-2020-2081_ref_011_w2aab3b7d174b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"P.  Pellizzari,\nEfficient Monte Carlo pricing of European options using mean value control variates,\nDecis. Econ. Finance 24 (2001), no. 2, 107\u2013126.","DOI":"10.1007\/s102030170002"},{"key":"2021053122092006445_j_mcma-2020-2081_ref_012_w2aab3b7d174b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"F.  Portier and J.  Segers,\nMonte Carlo integration with a growing number of control variates,\nJ. Appl. Probab. 56 (2019), no. 4, 1168\u20131186.","DOI":"10.1017\/jpr.2019.78"},{"key":"2021053122092006445_j_mcma-2020-2081_ref_013_w2aab3b7d174b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"I. M.  Sobol\u2019,\nGlobal sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates,\nMath. Comput. Simulation 55 (2001), 271\u2013280,","DOI":"10.1016\/S0378-4754(00)00270-6"},{"key":"2021053122092006445_j_mcma-2020-2081_ref_014_w2aab3b7d174b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"X.  Wang and K.-T.  Fang,\nThe effective dimension and quasi-Monte Carlo integration,\nJ. Complexity 19 (2003), no. 2, 101\u2013124.","DOI":"10.1016\/S0885-064X(03)00003-7"},{"key":"2021053122092006445_j_mcma-2020-2081_ref_015_w2aab3b7d174b1b6b1ab2ac15Aa","doi-asserted-by":"crossref","unstructured":"X.  Wang and I. H.  Sloan,\nWhy are high-dimensional finance problems often of low effective dimension?,\nSIAM J. Sci. Comput. 27 (2005), no. 1, 159\u2013183.","DOI":"10.1137\/S1064827503429429"}],"container-title":["Monte Carlo Methods and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/mcma\/ahead-of-print\/article-10.1515-mcma-2020-2081\/article-10.1515-mcma-2020-2081.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2020-2081\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2020-2081\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,1]],"date-time":"2021-06-01T01:21:08Z","timestamp":1622510468000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2020-2081\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,13]]},"references-count":15,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2,2]]},"published-print":{"date-parts":[[2021,6,1]]}},"alternative-id":["10.1515\/mcma-2020-2081"],"URL":"https:\/\/doi.org\/10.1515\/mcma-2020-2081","relation":{},"ISSN":["1569-3961","0929-9629"],"issn-type":[{"value":"1569-3961","type":"electronic"},{"value":"0929-9629","type":"print"}],"subject":[],"published":{"date-parts":[[2021,1,13]]}}}