{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:36:43Z","timestamp":1787341003758,"version":"build-2736575974"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Math."],"published-print":{"date-parts":[[2016,1]]},"abstract":"<jats:p>In this paper, we prove the existence of an integral closed-form solution for pricing barrier options in both Heston and Bates frameworks. The option value depends on time, on the price, and on the volatility of the underlying asset, and it can be computed as the solution of a two-dimensional pricing partial integro-differential equation. The integral representation formula of the solution is derived by projection of the differential equation and exploiting the properties of the adjoint operator. We derive the expression of the fundamental solution (Green's function) necessary for the integral representation formula. The computation is based on the interpretation of the fundamental solution as the joint transition probability density function of the underlying asset price and variance and is obtained through Fourier inverse transform of a suitable conditional characteristic function. We propose a numerical scheme to approximate the option price based on the classical boundary element method, and we provide two numerical examples showing the computational efficiency and accuracy of the proposed new method. The algorithm can be modified to compute greeks as well.<\/jats:p>","DOI":"10.1137\/15100504x","type":"journal-article","created":{"date-parts":[[2016,1,6]],"date-time":"2016-01-06T11:51:02Z","timestamp":1452081062000},"page":"27-57","source":"Crossref","is-referenced-by-count":21,"title":["Fast Numerical Pricing of Barrier Options under Stochastic Volatility and Jumps"],"prefix":"10.1137","volume":"76","author":[{"given":"C.","family":"Guardasoni","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S.","family":"Sanfelici","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2016,1,6]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s10614-013-9388-5"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"L.V. Ballestra and G. Pacelli,\n                      Pricing Double-Barrier Options Using the Boundary Element Method\n                      , Working paper, Universita\u0300 Politecnica delle Marche, Ancona, Italy, 2009. Available online at http:\/\/ssrn.com\/abstract=1492653 or http:\/\/dx.doi.org\/10.2139\/ssrn.1492653.","DOI":"10.2139\/ssrn.1492653"},{"key":"atypb3","first-page":"1","volume":"77","author":"Ballestra L.V.","year":"2014","journal-title":"Appl. Numer. Anal."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2011.09.050"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1093\/rfs\/9.1.69"},{"key":"atypb6","unstructured":"F. Bervoets,\n                      Model Risk for Exotic Equity Options\n                      , M.Sc. thesis, Applied Mathematics, Delft University of Technology, Delft, The Netherlands, 2006."},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1086\/260062"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.3905\/jod.1994.407891"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1287\/opre.1050.0247"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.21314\/JCF.1999.043"},{"key":"atypb11","unstructured":"G. Chen and J. Zhou,\n                      Boundary Element Methods\n                      , Academic Press, New York, 1992."},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.enganabound.2004.12.001"},{"key":"atypb13","unstructured":"R. Cont and P. 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Menaldi,\n                      Second Order Elliptic Integro-differential Problems\n                      , Chapman & Hall\/CRC, Boca Raton, FL, 2002.","DOI":"10.1201\/9781420035797"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"P. Glasserman,\n                      Monte Carlo Method in Financial Engineering\n                      , Springer, Berlin, 2004.","DOI":"10.1007\/978-0-387-21617-1"},{"key":"atypb21","doi-asserted-by":"publisher","unstructured":"C. Guardasoni and S. Sanfelici,\n                      A boundary element approach to barrier option pricing in Black-Scholes framework\n                      , Int. J. Comput. Math., to appear. DOI: 10.1080\/00207160.2015.1020304.","DOI":"10.1080\/00207160.2015.1020304"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1093\/rfs\/6.2.327"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/0304-405X(76)90022-2"},{"key":"atypb25","doi-asserted-by":"crossref","unstructured":"E. Miglio and C. Sgarra,\n                      A finite element discretization method for option pricing with the Bates model\n                      , S$\\vec{\\rm e}$MA J. (2011), pp. 23-40.","DOI":"10.1007\/BF03322591"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(99)00230-7"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-1889(00)00002-6"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.3905\/jod.1995.407939"},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"M. Rockinger and M. Semenova,\n                      Estimation of Jump-Diffusion Processes via Empirical Characteristic Functions\n                      , Research Paper 150, International Center for Financial Asset Management and Engineering, Swiss Finance Institute, Zurich, Switzerland, 2005. 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