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### Realised Volatility v Calculated Daily Volatility

Posted: January 2nd, 2020, 3:28 pm
if annualized volatility is 19.105% then daily volatility should be 1% assuming daily volatility is same across.

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).

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

### Re: Realised Volatility v Calculated Daily Volatility

Posted: January 3rd, 2020, 3:34 am
pankajchitlangia:

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.

### Re: Realised Volatility v Calculated Daily Volatility

Posted: January 3rd, 2020, 4:36 am
pankajchitlangia:

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.
Thanks Alan