1. Spot = $20
Strike = $18
Duration = 1 mo.
1 mo. Volatility = 5% x
S=$18
σ =-2
$19
$20
σ=-1
μ=0
Fig. 1. One-month $ and normalized σ spot price distribution.
One-month volatility = ($20-$19)/$20 = 0.05 or 5% which is normalized to =1σ by definition.
Strike price = $18, spot price = $20. The x axis is understood to be denominated in standard deviations, not dollars.
The area to the right of σ = -2 represents “in-the-money” at 1 mo.

2. Fig. 2. Three-month $ and normalized σ spot price distribution.
Strike = $18
Duration = 3 mo.
Volatility = 8.7%
In-the-money x
σ= -2
σ=-1
$20
S=$18
σ= -1.15
μ=0
Fig. 2. Three-month $ and normalized σ spot price distribution.
Three-month volatility = 5% x √3 = 8.7%
Strike price = $18, spot price = $20.
Note that the strike price has moved from -2σ to -2/√3 = -1.15σ in the 3 month distribution.
The area to the right of σ = represents “in-the-money” at 3 mos.

3. Spot = $20
Strike = $25
Duration = 3 mo.
1 mo. Volatility = 10%
3 mo. volatility = 17.3%
Out-of-the-money x
$20
$25
σ=1.44
μ=0
Fig. 3. Three-month $ and normalized σ spot price distribution.
Three-month volatility = 10% x √3 = 17.3%%
Strike price = $25, spot price = $20.
The strike price of $25 is at σ = ($25-$20)/2/1.73 = 1.44.
The area to the right of σ = 1.44 represents “in-the-money” at 3 mos.