How to calculate 1 Standard Deviation Move

I have seen much confusion about how to calculate 1 standard deviation move given the implied volatility of an option. First off, for reasons I will explain in another post, given a particular expiry you would want to use ATM implied volatility for that particular expiry.

All this starts with what exactly implied volatility is. True, that it is the backed out volatility from the BSM (Black-Scholes-Merton pricer. It is less known the Merton had independently come up with the Black-Scholes pricer and even received Nobel prize along with Black and Scholes for his options pricing work) that gives the current market option price. However, behind all this is the fact the BSM model assumes normally distributed returns. A normal distribution is parameterized by its mean and standard deviation. Implied volatility is the standard deviation of this distribution of returns. In the BSM model, time is measured in years so implied volatility is the annualized standard deviation of returns assuming a normal distribution of returns. There is the iid assumption in Black Scholes model which we will get into another post.

Bottom line is implied volatility is standard deviation of 1 year returns which has a normal distribution. If you quote implied volatility as 35% then you are saying that on 100 dollars invested the standard deviation of your returns is 35 dollars. Same number can be quoted as .35, however the interpretation is on 1 dollar invested the standard deviation of your return in one year is .35 dollars.

So note two things that when we have an ATM implied volatility such as .35, it is in essence 1 year standard deviation for 1 dollar invested.

Lets say currently the underlying is at 60 dollars and you would like to calculate standard deviation in 90 days time. Lets do this step by step.

1. Since 1 year s.d. on 1 dollar is .35, 1 year s.d. on 60 dollars is 60 * .35 = 21 dollars.

2. Now we have a number that gives us 1 year s.d. on 60 dollars but the time frame we have in mind is 90 days so we have to scale down 1 year s.d. of 21 dollars to 90 days. To do this one has to use the rule that for normal iid distributions s.d. scales with square root of time.

So we have to multiply the 1 year s.d. by sqrt(90/365).

Thus the answer is = .35 * stock price * sqrt(number of days/365)

This can be generatlized to:

standard deviation = stock price * ATM volatility * sqrt(number of days/365)

where number of days is the number of days out you want to calculate s.d. for.

There is related argument that stock prices change only on trading days and thus one should take into account only trading days since days when no trading is done do not contribute to volatility. If you are interested in calculation that takes this in account, you can use:

standard deviation = stock price * ATM volatility * sqrt(number of trading days to date you have in mind/252)

where 252 is roughly the number of trading days in year