An asymptotic expansion with push-down of Malliavin weights. (English) Zbl 1257.91052
Summary: We derive 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.
MSC:
91G60 | Numerical methods (including Monte Carlo methods) |
91G20 | Derivative securities (option pricing, hedging, etc.) |
60H07 | Stochastic calculus of variations and the Malliavin calculus |