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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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Research Papers in Economics

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- (2006). A.: Stochastic Calculus of Variations
- (1999). An Asymptotic Expansion Approach to Pricing Financial Contingent Claims,
- (1987). Analysis of Wiener Functionals (Malliavin Calculus) and its Applications to Heat Kernels,
- (1999). Applications of Malliavin calculus to MonteCarlo methods in Finance,
- (1992). Asymptotic Expansion for Small Diﬀusions via the Theory of Malliavin-Watanabe, Probability Theory and Related Fields, 92,
- (1992). Asymptotic Expansion for Statistics Related to Small Diﬀusions,
- (2009). Asymptotics of implied volatility in local volatility models: forthcoming in Mathematical Finance
- (2004). Computation in an Asymptotic Expansion Method, Preprint, CARF Working Paper Series CARF-F-149. (2009) 37[20]
- (2006). Computation of Greeks using Malliavin’s calculus in Jump type Market Models,
- (2000). Derivatives in ﬁnancial markets with stochastic volatility,
- (1995). Essays on the Valuation Problems of Contingent Claims,
- (2008). Further Developments in Volatility Derivatives Modeling, Global Derivatives Trading & Risk Management,
- (1984). Lectures on Stochastic Diﬀerential Equations and Malliavin Calculus,
- (1983). Malliavin’s calculus in terms of generalized Wiener functionals,
- (2002). Managing Smile Risk, Wilmott magazine,
- (2000). Option Valuation under Stochastic Volatility with Mathematica Code,
- (2008). P.H.: Analysis, Geometry and Modeling
- (2009). Pricing Options under Stochastic Volatility: a Power Series Approach, Finance Stoch.
- (2009). Second order expansion for implied volatility in two factor local stochastic volatility models and applications to the dynamic λ-SABR model,
- (1997). Small noise expansion and importance sampling, Asymptotic Anal.
- (1997). Stochastic Analysis,
- (2006). The Malliavin Calculus and Related Topics,
- Weak and strong Taylor methods for numerical solutions of stochastic diﬀerential equations, Quantitative Finance.(2010)

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