Black-Scholes on one page
Price = average payoff across plausible futures, weighted by likelihood, discounted. The formula computes that exactly.
Formula
C = S·N(d₁) − K·e−rT·N(d₂) P = K·e−rT·N(−d₂) − S·N(−d₁)
d₁ = [ln(S/K) + (r + σ²/2)·T] / (σ√T) d₂ = d₁ − σ√T
| Symbol | Means | Where it lives in the picture |
|---|---|---|
| S | Spot price today | Centre of the fan at day zero |
| K | Strike | The dashed line |
| T | Time to expiry, in years (days ÷ 365) | How far the fan has spread |
| σ (sigma, "IV") | Annual volatility as a decimal | How fast the fan spreads |
| r | Risk-free rate | Slight upward drift of the pile; discounting |
| N(d₂) | Probability of finishing in the money | Share of the pile beyond the strike |
| N(d₁) | Delta of a call; N(d₂) tilted up for "how far in" | Weighted share, counting the deep endings more |
What moves the price
| Input up | Call | Put | Picture |
|---|---|---|---|
| Spot | up | down | Pile slides right |
| Strike | down | up | Dashed line slides right |
| Days to expiry | up | up | Fan widens |
| Volatility | up | up | Fan widens faster |
| Rate | up | down | Pile drifts up slightly; strike discounted more |
Sanity anchors
At-the-money call, 30 days, 25% IV, no rate: premium is about 0.4 × S × σ × √T. For Reliance ₹2,500: 0.4 × 2500 × 0.25 × √(30/365) ≈ ₹72. Handy for spotting a mispriced quote.
Spread of the fan after T: roughly S × σ × √T. Reliance, 30 days, 25%: ± ₹180 covers about two-thirds of futures.
Where the model lies
Assumes symmetric log-returns, constant σ, no jumps, no costs. Reality crashes harder than it rallies, so put IVs exceed call IVs (skew). Traders use market price → IV, not IV → price. The model is a ruler.
Sources: Black & Scholes 1973 · Gundersen's intuitive derivation · Areal on N(d₁) and N(d₂) · Varsity Greek Calculator