Before revision:  We derive an almost sure limit and a large deviations result for the log-asset price in the n \infty limit . The rate function of the large deviations result does not depend on the discretization time step \tau. The implied volatility surface \sigma_{\rm BS for arbitrary maturity and strike in the limit \omega^2 T 0 , \sigma_0^2 T \infty at fixed (\omega^2 T)(\sigma_0^2 T) is represented as an extremal problem .
After revision:  We derive an almost sure limit and a large deviations result for the log-asset price in the n limit of large number of time steps. We derive an exact representation of the implied volatility surface \sigma_{\rm BS for arbitrary maturity and strike in the limit \omega^2 T 0 , \sigma_0^2 T \infty at fixed (\omega^2 T)(\sigma_0^2 T) is represented as an extremal problem .