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"An Asymptotic Expansion with Push-Down of Malliavin Weights"

By Akihiko Takahashi and Toshihiro Yamada

Abstract

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

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