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An Asymptotic Expansion with Push-Down of Malliavin Weights
en
Malliavin calculus
Asymptotic expansion
Stochastic volatility
Implied volatility
Local volatility
Shifted log-normal model
Jump-diffusion model
Integration-by-parts
Malliavin weight
Push-down
Bismut identity
Takahashi Akihiko
Yamada Toshihiro
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, which provides an expansion formula for generalized Wiener functionals and the closed-form approximation formulas in stochastic volatility environment. In addition, it presents applications of the general formula to a local volatility expansion in the stochastic volatility model and expansions of option prices in the shifted log-normal and jump-diffusion models with stochastic volatilities. Finally, with an application of the Bismut identity the paper shows an expansion of the solution to the partial differential equation for pricing in a stochastic volatility model.
Revised in January 2010, August 2010 and April 2011
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Discussion paper series. CIRJE-F
CIRJE-F-695
2009-12
AA11450569
application/pdf
335
日本経済国際共同センター
http://www.cirje.e.u-tokyo.ac.jp/research/dp/2009/2009cf695ab.html
http://hdl.handle.net/2261/50185