{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T00:05:25Z","timestamp":1759104325241},"reference-count":22,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,6,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The multilevel Monte Carlo (MLMC) method developed by M.\u2009B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607\u2013617] has a natural application to the evaluation of <jats:italic>nested<\/jats:italic> expectations <jats:inline-formula id=\"j_mcma-2020-2062_ineq_9999_w2aab3b7e2016b1b6b1aab1c14b1b3Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>\ud835\udd3c<\/m:mi>\n                              <m:mrow>\n                                 <m:mo>[<\/m:mo>\n                                 <m:mi>g<\/m:mi>\n                                 <m:mrow>\n                                    <m:mo>(<\/m:mo>\n                                    <m:mi>\ud835\udd3c<\/m:mi>\n                                    <m:mrow>\n                                       <m:mo>[<\/m:mo>\n                                       <m:mi>f<\/m:mi>\n                                       <m:mrow>\n                                          <m:mo>(<\/m:mo>\n                                          <m:mi>X<\/m:mi>\n                                          <m:mo>,<\/m:mo>\n                                          <m:mi>Y<\/m:mi>\n                                          <m:mo>)<\/m:mo>\n                                       <\/m:mrow>\n                                       <m:mo>|<\/m:mo>\n                                       <m:mi>X<\/m:mi>\n                                       <m:mo>]<\/m:mo>\n                                    <\/m:mrow>\n                                    <m:mo>)<\/m:mo>\n                                 <\/m:mrow>\n                                 <m:mo>]<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>{\\mathbb{E}[g(\\mathbb{E}[f(X,Y)|X])]}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, where <jats:inline-formula id=\"j_mcma-2020-2062_ineq_9998_w2aab3b7e2016b1b6b1aab1c14b1b5Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>f<\/m:mi>\n                              <m:mo>,<\/m:mo>\n                              <m:mi>g<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>{f,g}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> are functions and <jats:inline-formula id=\"j_mcma-2020-2062_ineq_9997_w2aab3b7e2016b1b6b1aab1c14b1b7Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mo>(<\/m:mo>\n                              <m:mi>X<\/m:mi>\n                              <m:mo>,<\/m:mo>\n                              <m:mi>Y<\/m:mi>\n                              <m:mo>)<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>{(X,Y)}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> a couple of independent random variables.\nApart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios.\nIn this work, we focus on the computation of initial margin.\nWe analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function <jats:italic>g<\/jats:italic> (which might fail to be everywhere differentiable).\nParallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.<\/jats:p>","DOI":"10.1515\/mcma-2020-2062","type":"journal-article","created":{"date-parts":[[2020,5,26]],"date-time":"2020-05-26T16:00:23Z","timestamp":1590508823000},"page":"131-161","source":"Crossref","is-referenced-by-count":3,"title":["Multilevel Monte Carlo methods and lower\u2013upper bounds in initial margin computations"],"prefix":"10.1515","volume":"26","author":[{"given":"Florian","family":"Bourgey","sequence":"first","affiliation":[{"name":"Centre de Math\u00e9matiques Appliqu\u00e9es (CMAP) , CNRS, Ecole Polytechnique , Institut Polytechnique de Paris , Route de Saclay, 91128 Palaiseau Cedex , France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefano","family":"De Marco","sequence":"additional","affiliation":[{"name":"Centre de Math\u00e9matiques Appliqu\u00e9es (CMAP) , CNRS, Ecole Polytechnique , Institut Polytechnique de Paris , Route de Saclay, 91128 Palaiseau Cedex , France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Emmanuel","family":"Gobet","sequence":"additional","affiliation":[{"name":"Centre de Math\u00e9matiques Appliqu\u00e9es (CMAP) , CNRS, Ecole Polytechnique , Institut Polytechnique de Paris , Route de Saclay, 91128 Palaiseau Cedex , France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexandre","family":"Zhou","sequence":"additional","affiliation":[{"name":"CERMICS (ENPC) , Universit\u00e9 Paris-Est , 77455 , Marne-la-Vall\u00e9e , France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2020,4,15]]},"reference":[{"key":"2023040101144791235_j_mcma-2020-2062_ref_001_w2aab3b7e2016b1b6b1ab2b1b1Aa","unstructured":"M.  Abramowitz and I. A.  Stegun,\nHandbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables,\nU.\u2009S. Department of Commerce, Washington, 1964."},{"key":"2023040101144791235_j_mcma-2020-2062_ref_002_w2aab3b7e2016b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"A.  Agarwal, S.  De Marco, E.  Gobet, J. G.  L\u00f3pez-Salas, F.  Noubiagain and A.  Zhou,\nNumerical approximations of McKean anticipative backward stochastic differential equations arising in initial margin requirements,\nESAIM Proc., 65 (2019), 1\u201326.","DOI":"10.1051\/proc\/201965001"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_003_w2aab3b7e2016b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"R.  Avikainen,\nOn irregular functionals of SDEs and the Euler scheme,\nFinance Stoch. 13 (2009), no. 3, 381\u2013401.","DOI":"10.1007\/s00780-009-0099-7"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_004_w2aab3b7e2016b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"M.  Broadie, Y.  Du and C. C.  Moallemi,\nRisk estimation via regression,\nOper. Res. 63 (2015), no. 5, 1077\u20131097.","DOI":"10.1287\/opre.2015.1419"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_005_w2aab3b7e2016b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"M.  Broadie and P.  Glasserman,\nEstimating security price derivatives using simulation,\nManagement Sci 42 (1996), no. 2, 269\u2013285.","DOI":"10.1287\/mnsc.42.2.269"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_006_w2aab3b7e2016b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"K.  Bujok, B. M.  Hambly and C.  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Res. 56 (2008), no. 3, 607\u2013617.","DOI":"10.1287\/opre.1070.0496"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_010_w2aab3b7e2016b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"M. B.  Giles,\nMultilevel Monte Carlo methods,\nActa Numer. 24 (2015), 259\u2013328.","DOI":"10.1017\/S096249291500001X"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_011_w2aab3b7e2016b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"M. B.  Giles and T.  Goda,\nDecision-making under uncertainty: Using MLMC for efficient estimation of EVPPI,\nStat. Comput. 29 (2019), no. 4, 739\u2013751.","DOI":"10.1007\/s11222-018-9835-1"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_012_w2aab3b7e2016b1b6b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"M. B.  Giles and A.-L.  Haji-Ali,\nMultilevel nested simulation for efficient risk estimation,\nSIAM\/ASA J. Uncertain. Quantif. 7 (2019), no. 2, 497\u2013525.","DOI":"10.1137\/18M1173186"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_013_w2aab3b7e2016b1b6b1ab2b1c13Aa","unstructured":"D.  Giorgi, V.  Lemaire and G.  Pag\u00e8s,\nWeak error for nested multilevel Monte Carlo, preprint (2018), https:\/\/arxiv.org\/abs\/1806.07627."},{"key":"2023040101144791235_j_mcma-2020-2062_ref_014_w2aab3b7e2016b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"E.  Gobet and A.  Makhlouf,\nThe tracking error rate of the delta-gamma hedging strategy,\nMath. Finance 22 (2012), no. 2, 277\u2013309.","DOI":"10.1111\/j.1467-9965.2010.00466.x"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_015_w2aab3b7e2016b1b6b1ab2b1c15Aa","doi-asserted-by":"crossref","unstructured":"R. D.  Gordon,\nValues of Mills\u2019 ratio of area to bounding ordinate and of the normal probability integral for large values of the argument,\nAnn. Math. Statistics 12 (1941), 364\u2013366.","DOI":"10.1214\/aoms\/1177731721"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_016_w2aab3b7e2016b1b6b1ab2b1c16Aa","doi-asserted-by":"crossref","unstructured":"M. B.  Gordy and S.  Juneja,\nNested simulation in portfolio risk measurement,\nManagement Sci. 56 (2010), no. 10, 1833\u20131848.","DOI":"10.1287\/mnsc.1100.1213"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_017_w2aab3b7e2016b1b6b1ab2b1c17Aa","doi-asserted-by":"crossref","unstructured":"I.  Guo and G.  Loeper,\nPricing bounds for volatility derivatives via duality and least squares Monte Carlo,\nJ. Optim. Theory Appl. 179 (2018), no. 2,598\u2013617.","DOI":"10.1007\/s10957-017-1168-2"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_018_w2aab3b7e2016b1b6b1ab2b1c18Aa","doi-asserted-by":"crossref","unstructured":"L.  Gy\u00f6rfi, M.  Kohler, A.  Krzy\u017cak and H.  Walk,\nA Distribution-free Theory of Nonparametric Regression,\nSpringer Ser. Statist.,\nSpringer, New York, 2002.","DOI":"10.1007\/b97848"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_019_w2aab3b7e2016b1b6b1ab2b1c19Aa","unstructured":"P.  Hall and C. C.  Heyde,\nMartingale Limit Theory and its Application,\nProbab. Math. Statist.,\nAcademic Press, New York, 1980."},{"key":"2023040101144791235_j_mcma-2020-2062_ref_020_w2aab3b7e2016b1b6b1ab2b1c20Aa","doi-asserted-by":"crossref","unstructured":"T.  Hastie, R.  Tibshirani and J.  Friedman,\nThe Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed.,\nSpringer Ser. Statist.,\nSpringer, New York, 2009.","DOI":"10.1007\/978-0-387-84858-7"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_021_w2aab3b7e2016b1b6b1ab2b1c21Aa","doi-asserted-by":"crossref","unstructured":"M. B.  Haugh and L.  Kogan,\nPricing American options: A duality approach,\nOper. Res. 52 (2004), no. 2, 258\u2013270.","DOI":"10.1287\/opre.1030.0070"},{"key":"2023040101144791235_j_mcma-2020-2062_ref_022_w2aab3b7e2016b1b6b1ab2b1c22Aa","doi-asserted-by":"crossref","unstructured":"P.  Henry-Labord\u00e8re,\nDeep primal-dual algorithm for BSDEs: Application of Machine Learning to CVA and IM, SSRN (2017), https:\/\/ssrn.com\/abstract=3071506.","DOI":"10.2139\/ssrn.3071506"}],"container-title":["Monte Carlo Methods and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/mcma\/26\/2\/article-p131.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2020-2062\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2020-2062\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T19:03:15Z","timestamp":1680375795000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2020-2062\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,15]]},"references-count":22,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,4,15]]},"published-print":{"date-parts":[[2020,6,1]]}},"alternative-id":["10.1515\/mcma-2020-2062"],"URL":"https:\/\/doi.org\/10.1515\/mcma-2020-2062","relation":{},"ISSN":["1569-3961","0929-9629"],"issn-type":[{"value":"1569-3961","type":"electronic"},{"value":"0929-9629","type":"print"}],"subject":[],"published":{"date-parts":[[2020,4,15]]}}}