Volatility is quoted as an annualized percentage, which is no use on its own when the question is whether a stock will hold a level over the next three weeks. This lesson turns the number into a range in dollars.
Scaling to the horizon
An annualized volatility describes one year. To scale it to a shorter period, multiply by the square root of the fraction of a year, because the spread of returns grows with the square root of time. The standard form uses calendar days:
Expected move = price × IV × sqrt(days / 365)
Take a $200 stock with 40% implied volatility and an option 30 days from expiration. The square root of 30/365 is 0.287. So 200 × 0.40 × 0.287 gives about $23, or 11.5% of the price. That is the one-standard-deviation move: if the option market has priced the distribution correctly, the stock finishes within $23 of $200 about two thirds of the time, and outside it about one third of the time. Two standard deviations, $46, covers about 95%.
The $200 stock at 40% implied volatility, 30 days out. The darker band is one standard deviation, about two outcomes in three; the lighter one is two. Drag days from 30 to 120 and the band doubles in width, not quadruples.
The shaded bands are the expected move made visible. The chart's bands lean slightly to the upside, because a price can rise without limit but cannot fall below zero; the formula's plus or minus $23 is the symmetric shortcut, and for horizons of a few months the two barely differ.
Two points about the formula. First, use the implied volatility of the expiration you care about, not the 30-day headline, if the two differ. Second, count calendar days to expiration, because the annualized figure is quoted on a 365-day year.
The square root matters more than it looks. Four times the time gives twice the move, not four times: a one-week move is about half a one-month move, and a one-year move is about 3.5 times a one-month one. Option prices follow the same curve.
An at-the-money straddle on a $100 stock against days to expiration, at 30% volatility. Hover at 30 days and then at 120: four times the time buys about twice the straddle. Drag volatility and the whole curve scales with it.
The straddle: the market's own number
The formula uses a model's idea of the distribution. There is a more direct reading. An at-the-money straddle is a call and a put at the same strike, the strike nearest the stock price. Whoever buys it makes money if the stock moves far enough in either direction to cover the two premiums, so its price is what the market charges for a move of unknown direction. Divide the straddle price by the stock price and you have the market's expected move as a percentage, with no model in the way.
On the $200 stock, the one-month at-the-money straddle might trade at about $18.30, or 9.1%. That is smaller than the $23 one-standard-deviation move, and the ratio is not an accident. The straddle pays out the size of the move on average, whatever it turns out to be, and for a bell-shaped distribution the average size of a move is about 0.8 of one standard deviation. So the straddle runs at roughly 0.8 times the formula, and the formula is roughly 1.25 times the straddle. Traders switch between the two constantly. When someone says a stock is "priced for a 9% move," they usually mean the straddle; when they say "a one-sigma move," they mean the formula.
Reading the range
Expected moves are most useful as breakevens and strike guides. A trader selling options wants strikes outside the expected move; a trader buying a straddle needs the stock to travel further than the straddle cost. Both start from the same two numbers you can now compute from any chain.
The maths: the square root of time and the 0.8 rule
If each day's return is independent, variances add. Over n days the variance is n times one day's, so the standard deviation is √n times one day's. That gives move = S × v × √(days / 365), where v is the annualized implied volatility.
For a normal distribution with standard deviation σ, the average size of a move, ignoring its sign, is σ × √(2/π), about 0.798σ. That is called the mean absolute deviation. A straddle at expiry pays the size of the move, so for short-dated options with small rates its fair price is close to 0.8 × S × v × √T. This is the at-the-money rule from Module 1, doubled: 0.4 for the call and 0.4 for the put.
Worked example: $200 stock, 40% IV, 30 days. √(30/365) is 0.287, so the one-sigma move is 200 × 0.40 × 0.287, about $22.94. Times 0.798 gives $18.30. The full pricing model, at a zero rate, gives $18.29 for the straddle.
Your turn
Open SPY's chain. You will compute the one-standard-deviation move from the at-the-money IV, then price the straddle from the mids, and compare the two. Check the ratio against the 0.8 rule before you move on.