The model behind 'probability of finishing above / below'.
Probabilities on this site are model-estimated, derived from current option-implied volatility under a lognormal price model. They are estimates of what current market pricing implies — not predictions, and not certainties of any kind.
Model
Given spot price S, at-the-money implied volatility σ for the chosen expiration, and time T in years (calendar days ÷ 365):
P(finish above K) = N(d₂), where d₂ = [ln(S∕K) − σ²T∕2] ÷ (σ√T)
N is the standard normal cumulative distribution. P(below) = 1 − P(above). The distribution curve drawn on stock pages is the corresponding lognormal density.
Assumptions (all of them matter)
- Zero drift and zero interest/dividend adjustment. At short horizons the effect is small; at long horizons it grows. This is a deliberate simplification, stated here rather than hidden.
- One volatility per expiration. We use ATM IV; real markets price a skew (out-of-the-money puts usually carry higher IV), so tail probabilities from this model differ from what a full smile-based model would say.
- Lognormal returns. Real distributions have fatter tails; extreme moves are more likely than the model implies.
Touch probabilities (reaching a level at any time before expiration) are approximated as roughly twice the finish-beyond probability, capped at 100% — a standard rule of thumb, labeled as an approximation wherever it appears.
Methodology version: em-1.0 (since August 2026). If the method changes, previously recorded outputs keep the version that produced them.