This is beyond basic but -In the rates world, when talking about normality and lognormality are these two unrelated discussions 1. Normalized vol means bp vol and lognormality mean percentage or proportional vol.2. Can we have normalized vol with assuming the underlying rate follows a lognormal distribution. In general when we say an underlying is lognormally or normally distributed (e.g. stock price or a swap rate) we are referring not to the change in underlying but the underlying itself.The distribution of the change in the underlying surely is ALWAYS normally distributed, i.e. equal probability of a move up or move down. Honestly would appreciate some help and patience, this uncertainty is really bothering me.. I have no clue regarding the equity world, but what are the common models used there for equities, is SABR used for equities and is a beta of 1 used often?

I think your confusion comes from a lack of appreciation of different time scales.Over a very small time interval we can say, for a stock price (and ignoring 'unchanged'), that probability_up = 0.5 + very small correctionprobability_down = 0.5 - same very small correctionSuch a statement is true in the real-world and compatible with all sorts of distributions over longer time periods, likea day, a week, etc. For stocks, over a day, the return distribution is actually quite wide-tailed (kurtosis >> 3)and so not close to either normal or lognormal looking. If you don't understand the point, study the binomial model for a lognormal variate as explained say in Hull,looking at very small times.

Some more inputs.Heston not SABR is the standard for equities in my knowledge (working in rates).One issue with SABR (beta=1) is that the underlying is not a martingale when the rho is positive. Continuing with SABR. From a SABR implied volatility you can extract both normal and lognormal implied volatilities (when they exists).

thanks.Zhu have to admit I'm very much on the practical side, so what do you mean by the underlying is not a martingale when the rho is positive? Also what are the applied consequences of that?I know SABRs output is simply the black vol (lognormal) which you stick into BS to give you price. What is the purpose of getting a nomal vol out of SABR?Alan thanks I have a lot of reading to do. My myopic view was that stock prices being floored at zero are broadly lognormal and isnt that the only distribution which prevents non-negative values?

QuoteOriginally posted by: gammaslideAlan thanks I have a lot of reading to do. My myopic view was that stock prices being floored at zero are broadly lognormal and isnt that the only distribution which prevents non-negative values?You're welcome. There are an infinite number of distributions which ensure non-negative values.Even if we restrict ourselves to:'named' distributions with continuous support on the whole half-line, here are some names (wikipedia)