Black–Scholes

what an option should be worth, and the odds it pays off · Options & probability — the math of “what are the odds?”

A Nobel-winning formula that estimates what an option should be worth — and the odds it finishes in the money — from a few facts about the stock. You don’t need the equation; you need what it’s built from and what it answers.

• Stock price & strike price — where the stock is now versus the price the option lets you buy (or sell) at.
• Time to expiration — how long the option has to work out.
• Volatility (σ) — how much the stock typically swings. This is the single most important input: a jumpy stock has a wider range of possible outcomes, so its options are worth more.
• Risk-free rate — what you could earn risk-free today (read from the live Treasury curve), used to discount the future (the same time-value-of-money idea as the DCF).

On every stock card the app uses it to answer two plain questions: (1) Where is this stock likely to be in a year? — it shows the expected ±1σ range, the band the stock lands in roughly two-thirds of the time. (2) What are the odds it finishes at or above the analyst target one year from now? — a probability, not a promise. Note the exact claim: it is the chance of being there at the horizon, not of touching the level at some point along the way — a stock that tags the target in March and fades by year-end counts as a miss.

These are risk-neutral estimates, not forecasts. Black–Scholes doesn’t know whether a stock will go up — it assumes an average drift and asks, “given how much this stock moves, what’s the spread of outcomes?” It tells you what would have to happen, not what will. That’s why the app labels it a model estimate, not a prediction.

expected 1-yr range = price × e^(drift ± σ)
drift = r − σ² ÷ 2
odds of finishing ≥ target at 1 yr = N(d₂)
d₂ = [ ln(price ÷ target) + (r − σ² ÷ 2) × T ] ÷ (σ × √T)

σ (sigma)

— the stock’s annualized volatility — how much it typically swings

r

— the risk-free rate (the current 2-yr Treasury)

T

— time to the horizon, in years (here, 1)

e^( )

— the exponential function — turns a growth rate into a price multiple

N( )

— the normal-distribution function — converts a distance into a probability 0–1

d₂

— how many standard deviations the target sits from the likely outcome

• Wikipedia — the Black–Scholes model (full math) ↗