2020-08-08T12:59:59Zhttps://repository.dl.itc.u-tokyo.ac.jp/?action=repository_oaipmhoai:repository.dl.itc.u-tokyo.ac.jp:000428242020-04-17T06:40:58Z00062:07433:0743400009:07435:07436
A Remark on Approximation of the Solutions to Partial Differential Equations in FinanceengMalliavin calculusBismut indentityIntegration-by-partsSemigroupAsymptotic expansionShort time asymptoticsHeat kernel expansionsDerivatives pricingStochastic volatilityLocal volatilitySABR modelλ-SABR modelsHeston modelhttp://hdl.handle.net/2261/51286Technical ReportTakahashi, AkihikoYamada, ToshihiroThis paper proposes a general approximation method for the solution to a second-order parabolic partial differential equation(PDE) 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 present two types of its applications, approximations of derivatives prices and short-time asymptotic expansions of the heat kernel. In particular, we provide approximate formulas for option prices under local and 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 March 2012; forthcoming in Recent Advances in Financial Engineering 2011.本文フィルはリンク先を参照のことDiscussion paper series. CIRJE-FCIRJE-F-8422012-02AA11450569application/pdf335日本経済国際共同センターhttp://www.cirje.e.u-tokyo.ac.jp/research/dp/2012/2012cf842ab.html2017-06-16