We propose a novel time discretization for the log-normal SABR model dS_t = \sigma_t S_t dW_t, d\sigma_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 n%DIFDELCMD < \to %%% \infty at fixed \beta =%DIFDELCMD < \frac12%%% \omega^2 n^2\tau, \rho = \sigma_0\sqrt{\tau . We derive an almost sure limit and a large deviations result for the log-asset price in the n%DIFDELCMD < \to %%% \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%DIFDELCMD < \to %%% 0 , \sigma_0^2 T%DIFDELCMD < \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 for small maturity and extreme strikes  .
We propose a novel time discretization for the log-normal SABR model (W_t,Z_t)=\varrho, } which is a popular stochastic volatility model that is widely used in financial practice. Our time discretization is a variant of the Euler-Maruyama scheme . We study its asymptotic properties in the limit of a large number of time steps %DIFDELCMD < \to %%% %DIFDELCMD < \frac12%%% under a certain asymptotic regime which includes the case of finite maturity, small vol-of-vol and large initial volatility with fixed product of vol-of-vol and initial volatility . We derive an almost sure limit and a large deviations result for the log-asset price in the %DIFDELCMD < \to %%% limit of large number of time steps. We derive an exact representation of the implied volatility surface for arbitrary maturity and strike in %DIFDELCMD < \to %%% %DIFDELCMD < \to %%% this regime . Using this representation we obtain analytical expansions of the implied volatility for small maturity and extreme strikes , which reproduce at leading order known asymptotic results for the continuous time model .
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