An Asymptotic Expansion with Push-Down of Malliavin Weights

Akihiko Takahashi and Toshihiro Yamada
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Akihiko Takahashi: Faculty of Economics, University of Tokyo
Toshihiro Yamada: Mitsubishi UFJ Trust Investment Technology Institute Co.,Ltd. (MTEC)

No CARF-F-256, CARF F-Series from Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo

Abstract: This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in multi-dimensional stochastic volatility models. In particular, the integration-byparts 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 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 stochastic volatility environment. Some numerical examples are also shown.

Pages: 31 pages
Date: 2011-10
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