Convergence to Black-Scholes for Ergodic Volatility Models

We study the eeect of stochastic volatility on option prices. In the fast-mean reversion model for stochastic volatility of 5], we show that there is a full asymptotic expansion for the option price, centered at the Black-Scholes price. We show, however, that this price does not converge in a strong sense to Black-Scholes as the mean-reversion rate increases. We also introduce a general (possibly non-Markovian) ergodic model and prove that, assuming decaying correlation between volatility and asset price, the option price strongly converges to Black-Scholes.