The premium plotted below isn't a slider you set by hand — it's computed on every input change from the actual Black–Scholes closed-form solution for a European option, the same formula used to sanity-check real option desks:
d1 = [ln(S/K) + (r + σ²/2)·T] / (σ·√T)
d2 = d1 − σ·√T
Call price = S·N(d1) − K·e^(−rT)·N(d2)
Put price = K·e^(−rT)·N(−d2) − S·N(−d1)
N(x) = standard normal CDF (erf-based, Abramowitz–Stegun 7.1.26)
At S=K=100, r=5%, σ=20%, T=1y this returns a call ≈ $10.45 and a put ≈ $5.57 — the textbook reference values for those inputs, which is what this engine reproduces (check it with the sliders above).
- Payoff pane (left) — the payoff at expiration,
max(S−K,0) − premium for a call or max(K−S,0) − premium for a put, using the live Black–Scholes premium. The dot marks the current spot on that curve.
- Price vs. volatility (top right) — the theoretical price recomputed across a range of σ, holding S, K, T, r fixed. Options get more valuable as volatility rises — more upside without more downside for the buyer — which is exactly what "vega" measures.
- Price vs. time (bottom right) — the theoretical price recomputed across time to expiry, holding everything else fixed. More time means more chance the option finishes in the money, so price falls toward intrinsic value as T→0 — the decay known as "theta."