A Remark on Approximation of the Solutions to Partial Differential Equations in Finance
34 Pages Posted: 19 Feb 2012
Date Written: February 19, 2012
Abstract
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 Leandre's approach (Leandre, 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 (lambda-) SABR models (Hagan et.al., 2002, Labordere, 2008) as special cases. Some numerical examples are shown.
Keywords: Malliavin calculus, Bismut indentity, Integration-by-parts, Semigroup, Asymptotic expansion, Short time asymptotics, Heat kernel expansions, Derivatives pricing, Stochastic volatility, Local volatility, SABR model, lambda-SABR models, Heston model
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