d1 and d2 in Black-Scholes
d1 and d2 are the two standardized distances between spot and strike that appear in the Black-Scholes formula. N(d2) is the model's risk-neutral probability that a call finishes in the money, and N(d1), adjusted for dividends, is the call's delta.
Senzoukria · Glossary · Updated September 2026
At a glance
- d1
- [ln(S/K) + (r − q + σ²/2)·τ] / (σ√τ)
- d2
- d1 − σ√τ
- Reading
- N(d2): risk-neutral probability of finishing in the money (call); e^(−qτ)·N(d1): call delta
Definitions
Both quantities measure how far the strike is from the forward price in units of standard deviation over the option's life. The numerator of d2, ln(S/K) + (r − q − σ²/2)·τ, is the expected log distance between the terminal price and the strike under the model's risk-neutral measure; dividing by σ√τ standardizes it. d1 is d2 shifted up by σ√τ. The shift comes from weighting outcomes by the terminal price itself, which is why N(d1) relates to delta rather than to a probability of exercise.
Worked example
Spot 100, strike 105, 30 days (τ = 0.0822), σ = 20%, r = q = 0. Then ln(100/105) = −0.0488 and σ√τ = 0.0573. d1 = (−0.0488 + 0.02 × 0.0822)/0.0573 = −0.822 and d2 = −0.822 − 0.057 = −0.880. N(d1) = 0.205, which is the call's delta, and N(d2) = 0.190, the model probability of the call finishing above 105. The call is worth 100 × 0.205 − 105 × 0.190 = 0.64.
| Quantity | Value | Meaning |
|---|---|---|
| d1 | −0.822 | Standardized distance, measure weighted by price |
| d2 | −0.880 | Standardized distance to the strike |
| N(d1) | 0.205 | Call delta (q = 0) |
| N(d2) | 0.190 | Risk-neutral probability of finishing in the money |
Limits of the probability reading
- N(d2) is a risk-neutral probability computed with the implied volatility of that strike. It is not a forecast of the real-world probability, which would require a view on expected return and on the true distribution.
- Delta is often quoted as a rough probability of finishing in the money; the example shows the two differ (0.205 against 0.190), more so for long maturities and high volatility.
- With a smile, the probability implied by the full set of prices includes a correction for how implied volatility changes with strike; using one strike's N(d2) ignores it.
Where d1 and d2 appear elsewhere
Every Black-Scholes greek is written with d1 and d2: gamma and vega through φ(d1), vanna through d2, charm through d2/(2τ), vomma through the product d1·d2. Senzoukria's GEX module uses d2 directly when it derives vanna and charm from each leg's gamma, spot, strike, implied volatility and time to expiry, and uses d1 in the Black-Scholes delta and gamma it computes locally for the gamma profile and the 25-delta skew.
Related
In the same section
- Daily loss limit
- D-shaped profile
- Price limits
- CVD anchor
- Settlement price
- Cumulative depth
- Conflation
- Cumulative delta
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Frequently asked questions
- Why is N(d1) not the probability of exercise?
- Because d1 includes an extra σ√τ that comes from measuring outcomes in units of the underlying rather than in cash. N(d1) is the probability of exercise under that share-based measure, which makes it the call's hedge ratio; the cash-measure probability of exercise is N(d2).
- What does a negative d2 tell me?
- That the strike sits above the model's risk-neutral median outcome for a call, so the call is more likely than not to expire worthless under the model. For an at-the-money option d2 is slightly negative because of the −σ²/2 term.