Asymptotic analysis for stochastic volatility: martingale expansion

A general class of stochastic volatility models with jumps is considered and an asymptotic expansion for European option prices around the Black-Scholes prices is validated in the light of Yoshida's martingale expansion theory. Several known formulas of regular and singular perturbation expansions are obtained as corollaries. An expansion formula for the Black-Scholes implied volatility is given which explains the volatility skew and term structure. The leading term of the expansion is always an affine function of log moneyness, while the term structure of the coefficients depends on the details of the underlying stochastic volatility model. Several specific models which represent various types of term structure are studied.