1> Analytical - Black Scholes formula for Vanilla European options, Digitals.
These valuations are just an "interpolation" of traded options. We interpolate the implied volatility from the traded points on the implied volatility surface.
There is no modeling assumption involved here.
Market uses this formula for implementing the "interpolation".
Scope for ...
You can use quadratic exponential method. For larger values of vol uses exponential function while for lower uses a quadratic ones, thus keeping the vol always positive.
This paper by Andersen as a reference to QE and other schemes too to apply to stochatic vol models (he refers to Heston, but you can adapt to SABR).