One standard deviation from the implied volatility, the probability it actually carries, the 252 against 365 trap, and why the straddle price is a smaller number.
Expected move = spot × implied volatility × √(days ÷ 365). On a stock at 100 with 24 per cent implied volatility, the 30-day one standard deviation move is 6.88, a range of 93.12 to 106.88, which the model expects to hold about 68 per cent of the time. The 30-day at-the-money straddle costs only 5.49, because a straddle prices the average absolute move, not one standard deviation.
Worked in full in Quantitative Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Implied volatility is quoted as an annual standard deviation of returns. The expected move converts it into the units a trader or risk manager actually uses: points on the underlying over a chosen horizon. The conversion is one line, but two conventions catch people out: the day count, and the difference between a standard deviation and the straddle price that many platforms label "expected move".
The underlying is the stock behind Kestrel 100, the worked option in Quantitative Finance. All figures are illustrative.
| Input | Value |
|---|---|
| Spot | 100.00 |
| Implied volatility, annualised on calendar days | 24% |
| Rate, continuous | 4% |
| Horizon | 30 days |
Variance grows in proportion to time, so standard deviation grows with its square root.
T = 30 ÷ 365 = 0.082192 years; √T = 0.2867
Expected move = S × σ × √T = 100 × 0.24 × 0.2867 = 6.88
Range: 100 ± 6.88 = 93.12 to 106.88; two standard deviations: 86.24 to 113.76
Excel: =S*IV*SQRT(Days/365)
As a percentage the 30-day move is 6.88 per cent of spot. For one day, √(1 ÷ 365) gives 100 × 0.24 ÷ 19.1050 = 1.2562: the same number as the gamma-theta break-even move, and not by coincidence, since a delta-hedged option breaks even when the stock moves by its implied daily standard deviation.
Under the lognormal model the range is not exactly symmetric in probability, but it is close. With the risk-neutral drift of 4 per cent, the chance of finishing between 93.12 and 106.88 is 68.29 per cent, against 68.27 for a normal distribution. The two standard deviation range holds 95.34 per cent, against 95.45. Strictly, a log-symmetric range, 100 × e±σ√T, would be 93.35 to 107.12; the up-move is larger than the down-move because prices compound.
A straddle, the at-the-money call plus the at-the-money put, pays the absolute move at expiry. Its price is therefore close to the expected absolute move, which for a normal distribution is √(2 ÷ π), or 0.7979, of one standard deviation.
30-day call 2.9072 + put 2.5789 = straddle 5.49 (Black-Scholes at 24%)
Mean absolute move: 6.88 × 0.7979 = 5.49
One standard deviation from the straddle: 5.49 ÷ 0.7979 = 6.88, or 1.25 times the straddle
Both numbers are useful. The straddle is what the market charges to own the move; the standard deviation is the edge of the 68 per cent range. Reading a 5.49 straddle as the one standard deviation move understates the range by a fifth.
| Days | One sd move | Range | ATM straddle | Probability in range |
|---|---|---|---|---|
| 1 | 1.26 | 98.74 to 101.26 | 1.00 | 68.3% |
| 7 | 3.32 | 96.68 to 103.32 | 2.65 | 68.3% |
| 30 | 6.88 | 93.12 to 106.88 | 5.49 | 68.3% |
| 91 | 11.98 | 88.02 to 111.98 | 9.54 | 68.3% |
| 365 | 24.00 | 76.00 to 124.00 | 18.99 | 68.5% |
Quadrupling the horizon only doubles the move: 91 days is 11.98, a year 24.00. Over a year the straddle's ratio to the move drifts down because of the interest rate, which lowers the put relative to the call. Volatility scales the move directly: at 16 per cent the 30-day move is 4.59, at 32 per cent it is 9.17.
Spot × implied volatility × √(days ÷ 365): 6.88 over 30 days at 24 per cent on a spot of 100, with a straddle of 5.49 sitting at 0.80 of it. Keep the day count consistent and remember which of the two numbers you are quoting. The workbook that prices Kestrel 100 across volatilities and maturities, with the Greeks checked by finite differences, is on the free companion page for this book.
It is the expected absolute move, not the one standard deviation move. For a normal distribution the mean absolute deviation is root(2/pi), or 0.7979, of a standard deviation. At 100 and 24 per cent over 30 days the straddle costs 5.49 and the one standard deviation move is 6.88, so 1.25 times the straddle.
Match the volatility's convention. Option implied volatilities are normally annualised on calendar time, so use days to expiry over 365: 1.2562 a day at 24 per cent on a spot of 100. Using trading days, a day is 1/252 of a year and the daily move is 1.5119. Mixing them misstates the range.
About 68 per cent for one standard deviation and about 95 per cent for two, if returns are lognormal at the implied volatility. At 24 per cent over 30 days the probability of finishing between 93.12 and 106.88 is 68.29 per cent; between 86.24 and 113.76 it is 95.34 per cent. Real returns have fatter tails, so treat these as model figures.
This article is one calculation from Quantitative Finance. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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