Ask a trader how much a stock moved today and you will hear a percentage, not a dollar figure. A $2 move is huge for a $20 stock and noise for a $2,000 one. Pricing models think the same way. They describe the stock's future as a spread of percentage moves, and much of what follows in this course comes from that one choice.
Why percent
Three reasons. First, a percentage move means the same thing at any price level, so a single volatility number can describe a stock whether it trades at $50 or $500. Second, a stock cannot go below zero, but there is no ceiling above it. A model that dealt in dollar moves would happily let a $30 stock fall by $40. Percentages fix that: a 50% fall from $30 is $15, a further 50% fall is $7.50, and the price never reaches zero however many falls you stack. Third, percentage moves compound. A stock that rises 10% and then falls 10% does not end where it started. $100 becomes $110, then $99.
Put those together and the spread of possible future prices is lopsided in dollars. A $100 stock that could halve could also double, and those two outcomes are $50 down and $100 up. Measured in compounding percentage terms, called log returns, the picture is a symmetric bell curve. In dollar terms it has a short left side and a long right tail. Statisticians call that shape lognormal, and it is the picture inside the standard model: log returns are bell-shaped, so prices are lognormal.
A $100 stock at 40% volatility with 30 days left. Drag days from 30 to 120 and the bands only double in width: the square root of four is two. Push volatility up and the right tail stretches further than the left.
Read the note under the chart. At 40% volatility and 30 days, the one-standard-deviation range on a $100 stock runs from about $89 to about $112: $11 down, $12 up, the same size in log terms either way.
What annualized volatility means
Volatility is quoted as a yearly figure. A volatility of 40% means that over one year a move of about 40%, up or down, is a one-standard-deviation move. Roughly two years in three the stock finishes within that range. That is the whole definition. It says nothing about direction, and it is not a maximum.
Shrinking it to the option's life
Nobody holds a one-month option for a year, so the yearly figure has to be scaled to the days the option actually has. Moves do not add up in a straight line, because some of them cancel each other out. What adds up in a straight line is the variance, the square of the typical move, so the typical move itself grows with the square root of time. To get the one-standard-deviation move over any period, multiply the annual volatility by the square root of that period as a fraction of a year.
Take a $120 stock at 40% volatility with 30 days to expiration. The square root of 30/365 is 0.287. So the 30-day move is 40 times 0.287, about 11.5%, which on $120 is about $13.80. Two times in three the stock is expected to finish between roughly $106 and $134.
- One month is about 1/12 of a year, and the square root of 1/12 is 0.29. Divide the annual vol by about 3.5 to get a monthly move.
- One trading day is 1/252 of a year, and the square root is 0.063. Divide by about 16 for a daily move. A 40 vol stock moves about 2.5% on a typical day.
This rule is also why a 90-day option costs about 1.7 times a 30-day one rather than three times: the expected move grows with the square root of three, not three. Drag the days slider on the chart from 30 to 120: four times the days, and the shaded band only doubles in width.
Why the typical outcome ends below today
Here is the strangest consequence of thinking in percentages. If up and down moves of the same size are equally likely, the middle outcome, the median, still ends below where the stock started. The reason is the compounding from the first section. Up 10% then down 10% loses 1%. Over many steps those small losses add up, and the drag grows with the square of volatility: half of volatility squared per year. At 60% volatility that is 18% a year, so with no drift the median one-year outcome for a $100 stock is about $84, even though the average outcome is still $100. The average is held up by the long right tail: a few huge winners balance many modest losers.
The model's chance that an at-the-money call finishes in the money, over two years, at three volatilities. Set the rate to 0% and every line falls below 50%: the median outcome sits below today, and further below the higher the volatility.
The chart shows the same effect from the option side. An at-the-money call finishes in the money only if the stock ends above today's price. At low volatility that is near a coin flip; at high volatility over a long time it falls well below half. Volatile stocks usually end lower and make up for it with the occasional very large gain.
The maths: lognormal returns and the median drag
The model says the log return over time T, ln(S_T / S), is normally distributed with mean (r - q - v²/2) T and standard deviation v √T. The -v²/2 term is the compounding drag. It is exactly what is needed to keep the average price growing at r - q: for a normal variable x, the average of e^x is e^(mean + sd²/2), and the two halves cancel.
So the median price is S e^((r - q - v²/2) T) while the mean is S e^((r - q) T). With r = q = 0, v = 0.60 and T = 1, the median is 100 × e^(-0.18) = $83.53 and the mean is $100.
The same picture shows why a 10% rise and a 10% fall are not mirror images. In log terms a 10% rise is ln(1.10) = 0.0953 and a 10% fall is ln(0.90) = -0.1054: the fall is the bigger step. At 40% volatility over 30 days, v √T = 0.1147, so the rise is 0.83 standard deviations away and the fall 0.92. Ignoring the small drift, the chance of finishing 10% or more higher is about 20%, and 10% or more lower about 18%. Equal percentages, unequal odds, which is the lognormal shape in two numbers.
Your turn
Open NVDA. Read the at-the-money IV off the chain, scale it to the expiration's days with the square-root rule, then turn the percentage into dollars using the price in the quote bar. That is the expected move, and you have just done what the model does.