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A Remark on Approximation of the Solutions to Partial Differential Equations in Finance
en
Malliavin calculus
Bismut indentity
Integration-by-parts
Semigroup
Asymptotic expansion
Short time asymptotics
Heat kernel expansions
Derivatives pricing
Stochastic volatility
Local volatility
SABR model
λ-SABR models
Heston model
Takahashi Akihiko
Yamada Toshihiro
This 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.
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Discussion paper series. CIRJE-F
CIRJE-F-842
2012-02
AA11450569
application/pdf
335
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http://www.cirje.e.u-tokyo.ac.jp/research/dp/2012/2012cf842ab.html