{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:25:57Z","timestamp":1787322357586,"version":"3.56.0"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Finan. Math."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>We present new extensions to a method for constructing families of solvable one-dimensional time-homogeneous diffusions whose transition densities and other related quantities are obtainable in analytically closed form. Our approach is based on a dual application of the so-called diffusion canonical transformation method that combines a smooth monotonic differentiable map with nonzero derivative (diffeomorphism) and a measure change via a Doob-$h$ transform. This gives rise to new multiparameter solvable diffusions that are divided into two main classes; the first is specified by having affine (linear) drift with various resulting nonlinear diffusion coefficient functions, while the second class allows for several specifications of a (generally nonlinear) diffusion coefficient with resulting nonlinear drift function. The first class of models, having linear drift and nonlinear (state-dependent) volatility functions, is useful for pricing equity and foreign exchange (FX) options in finance, while the second class of diffusions contains new models that are mean-reverting and are applicable to pricing interest-rate and other path-dependent derivatives such as volatility index (VIX) options. As specific examples of the first class of affine drift models, we present explicit results for two new families of models that arise from the squared Bessel process (the Bessel family) and the Ornstein--Uhlenbeck diffusion (the OU family). For the second class of nonlinear drift models, we give examples of solvable subfamilies called the Bessel family of mean-reverting diffusions and derive closed-form integral formulas for conditional expectations of certain functionals. In particular, we derive a new closed-form analytical formula for the Laplace transform with respect to the strike price of a standard call VIX option. We then succeed in Laplace inverting the expression to obtain numerically exact VIX call option prices with realistic implied volatilities with respect to strike and maturity. Moreover, we accurately calibrate the OU family of models to FX option market data, exhibiting pronounced implied volatility smiles across several strikes and maturities.<\/jats:p>","DOI":"10.1137\/15m1033502","type":"journal-article","created":{"date-parts":[[2017,2,15]],"date-time":"2017-02-15T13:21:41Z","timestamp":1487164901000},"page":"146-170","source":"Crossref","is-referenced-by-count":5,"title":["Solvable Diffusion Models with Linear and Mean-Reverting Nonlinear Drifts"],"prefix":"10.1137","volume":"8","author":[{"given":"Giuseppe","family":"Campolieti","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Roman N.","family":"Makarov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,2,15]]},"reference":[{"key":"atypb1","unstructured":"C. Albanese and G. Campolieti,\n                      Advanced Derivatives Pricing and Risk Management: Theory, Tools and Hands-On Programming Application\n                      , Academic Press, New York, 2006."},{"key":"atypb2","first-page":"99","volume":"14","author":"Albanese C.","year":"2001","journal-title":"Risk"},{"key":"atypb3","first-page":"563","volume":"15","author":"Albanese C.","year":"2009","journal-title":"Markov Process. Relat. Fields"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4149(78)90036-4"},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"A. N. Borodin and P. Salminen,\n                      Handbook of Brownian Motion-Facts and Formulae\n                      , 2nd ed., Probability Appl., Birkha\u0308user, Basel, 2002.","DOI":"10.1007\/978-3-0348-8163-0"},{"key":"atypb6","first-page":"2028285","author":"Branger N.","year":"2016","journal-title":"SSRN"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1142\/S0219024907004081"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1142\/S0219024913500271"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1080\/14697680701282186"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9965.2012.00521.x"},{"key":"atypb12","first-page":"121","author":"Campolieti G.","year":"2015","journal-title":"New York"},{"key":"atypb13","unstructured":"G. Campolieti, R. N. Makarov, and A. Vasiliev,\n                      Bridge Copula Model for Option Pricing\n                      , preprint,arXiv:1110.4669, 2011."},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2006.05.025"},{"key":"atypb15","unstructured":"J. Hakala and U. Wystup,\n                      Foreign Exchange Risk; Models, Instruments and Strategies\n                      , Risk Books, London, 2002."},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1080\/14697680601103045"},{"key":"atypb17","unstructured":"K. Ito and H. McKean, Jr.\n                      Diffusion Processes and Their Sample Paths\n                      , Springer-Verlag, Berlin, 1965."},{"key":"atypb18","unstructured":"S. Karlin and H. E. Taylor,\n                      A Second Course in Stochastic Processes\n                      , Academic Press, New York, 1981."},{"key":"atypb19","unstructured":"A. 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Sepp,\n                      VIX option pricing in a jump-diffusion model\n                      , Risk, (2008), pp. 84-89."},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2015.06.024"}],"container-title":["SIAM Journal on Financial Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/15M1033502","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:21:01Z","timestamp":1787318461000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/15M1033502"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/15M1033502"],"URL":"https:\/\/doi.org\/10.1137\/15m1033502","relation":{},"ISSN":["1945-497X"],"issn-type":[{"value":"1945-497X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}