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.
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