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SIAM J. on Financial Mathematics / Volume 3

SIAM J. Finan. Math. 3, pp. 95-136 (42 pages)

An Asymptotic Expansion with Push-Down of Malliavin Weights

Akihiko Takahashi and Toshihiro Yamada

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This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in multidimensional stochastic volatility models. In particular, the integration-by-parts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied. We provide an expansion formula for generalized Wiener functionals and closed-form approximation formulas in the stochastic volatility environment. In addition, we present applications of the general formula to expansions of option prices for the shifted log-normal model with stochastic volatility. Moreover, with some results of Malliavin calculus in jump-type models, we derive an approximation formula for the jump-diffusion model in the stochastic volatility environment. Some numerical examples are also shown.

© 2012 Society for Industrial and Applied Mathematics

KEYWORDS

AMS Subject Headings

60H07, 91G20, 91G60

PUBLICATION DATA

ISSN

1945-497X (online)

Publisher

$abstract.society.value is a Member of CrossRef Society for Industrial and Applied Mathematics

ARTICLE DATA

History
Received September 07, 2010
Accepted October 26, 2011
Published online January 24, 2012
Digital Object Identifier
http://dx.doi.org/10.1137/100807624

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