Asymptotic Methods for Computing Implied Volatilities under Stochastic Volatility

In this paper we propose analytical approximations for computing implied volatilities when time-to-maturity t is small. The analysis is performed in the framework of a two-factor model with local and stochastic volatility. We describe an algorithm for building the power series approximation of implied volatility. In the case of CEV volatility of volatility we first obtain a quasianalytical solution for the limit of implied volatilities Y as t --> 0. Then we show that implied volatilities of short term options can be accurately computed by a proper transformation of Y. We introduce a class of models for which this method may be accurate also for t >> 0. In the particular case of SABR model we obtain an extension of the formula derived in Hagan et al. (2002).