An Asymptotic Expansion with Push-Down of Malliavin Weights

作者：Takahashi Akihiko^*; Yamada Toshihiro

来源：SIAM Journal on Financial Mathematics, 2012, 3(1): 95-136.

DOI：10.1137/100807624

• Malliavin calculus
• asymptotic expansion
• stochastic volatility
• implied volatility
• shifted log-normal model
• jump-diffusion model
• integration-by-parts
• Malliavin weight
• push-down
• Malliavin calculus for Poisson processes

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.

• 出版日期2012