2022-01-19T13:57:49Zhttps://repository.dl.itc.u-tokyo.ac.jp/oaioai:repository.dl.itc.u-tokyo.ac.jp:000428112021-03-02T02:07:25ZOn Approximation of the Solutions to Partial Differential Equations in FinanceTakahashi, Akihiko98433Yamada, Toshihiro98434335Barrier OptionsKnock-out optionsSABR modelλ-)SABR modelsHeston modelShort time asymptoticsHeat kernel expansionsMalliavin calculusBismut indentityStochastic volatilityLocal volatilityIntegration-by-partsSemigroupDerivatives pricingapplication/pdfThis paper proposes a general approximation method for the solutions to second-order parabolic partial differential equations (PDEs) widely used in finance through an extension of Léandre's approach(Léandre (2006,2008)) and the Bismut identiy(e.g. chapter IX-7 of Malliavin (1997)) in Malliavin calculus. We show two types of its applications, new approximations of derivatives prices and short-time asymptotic expansions of the heat kernel. In particular, we provide new approximation formulas for plain-vanilla and barrier option prices under stochastic volatility models. We also derive short-time asymptotic expansions of the heat kernel under general time-homogenous local volatility and local-stochastic volatility models in finance which include Heston (Heston (1993)) and (λ-)SABR models (Hagan et.al. (2002), Labordere (2008)) as special cases. Some numerical examples are shown.Revised in January 2012.本文フィルはリンク先を参照のことtechnical report日本経済国際共同センター2011-08Discussion paper series. CIRJE-FCIRJE-F-815AA11450569enghttp://www.cirje.e.u-tokyo.ac.jp/research/dp/2011/2011cf815ab.htmlmetadata only access