We propose a novel time discretization for the log-normal SABR model $dS_t = sigma_t S_t dW_t, dsigma_t = omega sigma_t dZ_t$, with $mbox{corr}(W_t,Z_t)=varrho$, which is a variant of the Euler-Maruyama scheme, and study its asymptotic properties in the limit of a large number of time steps $nto infty$ at fixed $beta = frac12omega^2 n^2tau,rho = sigma_0sqrt{tau}$. We derive an almost sure limit and a large deviations result for the log-asset price in the $nto infty$ limit. The rate function of the large deviations result does not depend on the discretization time step $tau$. The implied volatility surface $sigma_{rm BS}(K,T)$ for arbitrary maturity and strike in the limit $omega^2 T to 0 , sigma_0^2 T to infty$ at fixed $(omega^2 T)(sigma_0^2 T)$ is represented as an extremal problem. Using this representation we obtain analytical expansions of $sigma_{rm BS}(K,T)$ for small maturity and extreme strikes.
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