d₁ & d₂ on the normal curve
d₁ and d₂ are the two points fed into N(·) in the Black-Scholes formula. Move the inputs and watch them slide along the standard normal curve. The blue area up to d₂ is N(d₂), the risk-neutral probability the call ends in the money. Add the gold strip from d₂ to d₁ and the whole shaded area becomes N(d₁), the call's delta — so N(d₁) = blue + gold.
N(d₂) = P(ITM), blue 0.528N(d₁) = delta, blue+gold 0.584
d₁0.2121
d₂0.0707
N(d₁)0.584
N(d₂)0.528
d₂ = d₁ − σ√T, so d₂ always sits to the left of d₁ by the total volatility σ√T.
Spot S100
Strike K100
Volatility σ20%
Maturity T0.50 yr
Rate r4.0%