On Approximation of the Solutions to Partial Differential Equations in Finance
AbstractThis 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.
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Bibliographic InfoPaper provided by CIRJE, Faculty of Economics, University of Tokyo in its series CIRJE F-Series with number CIRJE-F-815.
Length: 44 pages
Date of creation: Aug 2011
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- Devadoss, Stephen & Li, Shuangling & Jones, James, 1997. "A Stochastic simulation analysis of the effects of Indian wheat production instability on the world wheat market," Journal of Policy Modeling, Elsevier, vol. 19(4), pages 375-379, August.
- Akihiko Takahashi, 2006. "``On an Asymptotic Expansion Approach to Numerical Problems in Finance" (forthcoming in ``Sugaku," Vol. 59-1, (2007))," CARF J-Series CARF-J-032, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Dec 2009.
- Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
- Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
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