CIRJE-F-695 "An Asymptotic Expansion with Push-Down of Malliavin Weights"
Author Name Takahashi, Akihiko and Toshihiro Yamada
Date December 2009
Full Paper  
Remarks @Revised in January 2010, August 2010 and April 2011; revised as CIRJE-F-824 (2011); subsequently published in SIAM Journal on Financial Mathematics Volume 3, pp 95-136 (2012).

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.