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To accommodate these, we introduce the class of modulated Volterra processes and show that they successfully capture the VIX smile.<\/jats:p>","DOI":"10.1137\/18m1169242","type":"journal-article","created":{"date-parts":[[2020,4,27]],"date-time":"2020-04-27T14:16:57Z","timestamp":1587997017000},"page":"437-469","source":"Crossref","is-referenced-by-count":36,"title":["Volatility Options in Rough Volatility Models"],"prefix":"10.1137","volume":"11","author":[{"given":"Blanka","family":"Horvath","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3986-3201","authenticated-orcid":true,"given":"Antoine","family":"Jacquier","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter","family":"Tankov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2020,4,27]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1214\/19-AAP1477"},{"key":"atypb2","unstructured":"R. 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B\u00fchler,\n                      Volatility Markets: Consistent Modeling, Hedging, and Practical Implementation of variance Swap Market Models\n                      , Ph.D. thesis, Technische Universit\u00e4t Berlin, 2006."},{"key":"atypb14","unstructured":"P. Carr and D. Madan,\n                      Towards a theory of volatility trading\n                      , Risk (1998), pp. 417-427."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1080\/0740817X.2013.857063"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"G. Da Prato and J. Zabczyk,\n                      Stochastic equations in Infinite Dimensions\n                      , 2nd ed., Cambridge University Press, Cambridge, MA, 2014.","DOI":"10.1017\/CBO9781107295513"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1060202833"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-018-0360-z"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1111\/mafi.12173"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1214\/18-AAP1408"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1080\/14697688.2016.1197410"},{"key":"atypb22","unstructured":"M. Fukasawa,\n                      Hedging and calibration for log-normal rough volatility models\n                      , Presented at Jim Gatheral's 60th Birthday Conference, New York, 2017."},{"key":"atypb23","unstructured":"J. Gatheral,\n                      Consistent modelling of SPX and VIX options\n                      , Presented at Bachelier Congress, London, 2008."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1080\/14697688.2017.1393551"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-019-00392-5"},{"key":"atypb26","unstructured":"I. S. Gradshteyn and I. M. Ryzhik,\n                      Table of Integrals, Series, and Products\n                      , Academic Press, New York, 2014."},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1137\/17M1142892"},{"key":"atypb28","doi-asserted-by":"crossref","unstructured":"J. Guyon,\n                      The Joint S&P 500\/VIX Smile Calibration Puzzle Solved\n                      ,SSRN:3397382, 2019.","DOI":"10.2139\/ssrn.3397382"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-017-0334-6"},{"key":"atypb30","doi-asserted-by":"crossref","first-page":"25","DOI":"10.21314\/JCF.2018.361","volume":"22","author":"Hendriks S.","year":"2019","journal-title":"J. Comput. Finance"},{"key":"atypb31","doi-asserted-by":"crossref","unstructured":"T. Hida and M. Hitsuda,\n                      Gaussian Processes\n                      , AMS, Providence, RI, 1993.","DOI":"10.1007\/978-94-017-3680-0_1"},{"key":"atypb32","first-page":"56","author":"Horvath B.","year":"2019","journal-title":"J. Appl. Probab."},{"key":"atypb33","doi-asserted-by":"crossref","unstructured":"B. Horvath, A. Jacquier, and A. Muguruza,\n                      Functional Central Limit Theorems for Rough Volatility\n                      ,https:\/\/arxiv.org\/abs\/1711.03078, 2017.","DOI":"10.2139\/ssrn.3078743"},{"key":"atypb34","unstructured":"J. Jacod and A. Shiryaev,\n                      Limit theorems for Stochastic Processes\n                      , Grundlehren Math. Wiss., 288, Springer, New York, 2013."},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1080\/14697688.2017.1353127"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2009.10.012"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1016\/0378-4266(90)90039-5"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1137\/1010093"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1080\/14697688.2018.1459812"},{"key":"atypb40","first-page":"87","volume":"21","author":"Ould Aly S. M.","year":"2013","journal-title":"Appl. Math. Finance"},{"key":"atypb41","unstructured":"P. Protter,\n                      Stochastic Integration and Differential Equations\n                      , Springer, New York, 2014."}],"container-title":["SIAM Journal on Financial Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1169242","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:54:01Z","timestamp":1787320441000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1169242"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":41,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["10.1137\/18M1169242"],"URL":"https:\/\/doi.org\/10.1137\/18m1169242","relation":{},"ISSN":["1945-497X"],"issn-type":[{"value":"1945-497X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1]]}}}