if annualized volatility is 19.105% then daily volatility should be 1% assuming daily volatility is same across.
>> This is true only in an ideal world in which (1) stock prices follow geometric Brownian motion (GBM) with a constant volatility, and (2) realized volatility is measured by extremely fine time-interval sampling.
However, if I have an ATM options with implied vols of 19.105%, then if the futures prices has daily movement of 1% till expiry, why doesn't the realized volatility is equal to the premium paid (implied volatility).
>> Because the real world is different from that ideal GBM world. Sampling issues aside, there are other things going on. For example, 30-day SPX options tend to have an (annualized) implied volatility about 3 vol points higher than (average) realized 30-day volatility. The difference is largely explainable by simple risk aversion and hedging/insurance arguments.
I see that premium lost is equal to realized volatility till 3-4 days prior to expiry and start deviating from then till expiry. If someone can explain why is that please, thanks
>> Close to expiration is probably better described by a pure jump process than GBM. The implied volatility smile is tending more toward a "V" shape. A consequence is that, if you are looking at an option just slight out-of-the-money, it will seem to lose its premium unusually slowly relative to GBM.