Hi, For anybody interested in options out there… If you take the implied volatilty of an option and divide it by SQRT(252) you get the daily standard deviation change priced into the option. For example, an option priced with 16% annual volatility will have 1% change in underlying priced in for 2/3 trading days. I am trying to understand the relationship between the way vol is calculated and this transformation. As volatility (historical or implied) is a measure of the deviation of the log returns of the underlying and not the price changes of the underlying itself then I would have thought that doing the above transformation would give you the expected daily standard deviation of the log returns on the stock (and not the expected price change). E.g. a 16% annual implied vol would translate into a 1% variation of the expected log return on the stock. Could this mean that a stock that is expected to return 5% on day 1, 6.05% day two, 4.9995% on day three etc would be properley priced by selling an option with 16% implied vol? (as the returns are varying by 1% each day) My intuition tells me that it wouldn’t, I would expect to relaise a profit if I had bought the option priced at 16% When looking into this I calculated the historical vol for different price series. Interestingly, a stock that moves up exactly the same amount or down the same amount each day will have a historical vol of 0. It could move 100% up every day and still be 0!
I have figured it out…it is due to the forward being the expected price movement…as it is almost nothing on a daily basis it is OK to use this estimation
I’d be interested in hearing more about this, can I get your email address?
The reason the estimation can work is that the expected return (e.g. the forward rate: risk free rate - borrowing cost of stock) is insignificant on a daily basis. So a deviation of 1% from the expected return is very close to a 1% return in the stock that day e.g a 1% price change. Taleb discusses this in Dynamic Hedging around page 90 in the section “Calculating historical vol and correlation” you can see it on Amazon via LookInside.