An Asymptotic Expansion with Push-Down of Malliavin Weights

Akihiko Takahashi · Toshihiro Yamada

82 citations0/0 files storedSIAM Journal on Financial Mathematics

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Stochastic processes and financial applications Financial Risk and Volatility Modeling Fuzzy Systems and Optimization

Abstract

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.

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References

1983Stochastic Differential Equations and Diffusion Processes 2006The Malliavin Calculus and Related Topics 1997Stochastic Analysis 2000Derivatives in Financial Markets with Stochastic Volatility 1999Applications of Malliavin calculus to Monte Carlo methods in finance 1984Lectures on Stochastic Differential Equations and Malliavin Calculus 1987Analysis of Wiener Functionals (Malliavin Calculus) and its Applications to Heat Kernels 2002Option valuation under stochastic volatility: with mathematica code

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2020Implied Stochastic Volatility Models 2020Closed-form implied volatility surfaces for stochastic volatility models with jumps 2013Closed-Form Expansion, Conditional Expectation, and Option Valuation n.d.A weak approximation with asymptotic expansion and multidimensional Malliavin weights (Revised version of CARF-F-335; Forthcoming in Annals of Applied Probability")"2019An Arbitrary High Order Weak Approximation of SDE and Malliavin Monte Carlo: Analysis of Probability Distribution Functions

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