Black-Scholes Explorer

Inputs

What each input does

S — Stock price

Where the underlying trades now. Higher S lifts calls and weighs on puts.

K — Strike

The price you’d transact at. The gap between S and K is your moneyness.

T — Time (years)

More time means more chances to move — so more time value, for calls and puts alike.

r — Risk-free rate

The interest backbone. It nudges prices through the discount factor; usually a minor effect.

σ — Volatility

The single biggest price lever. More expected movement = more expensive options, both sides.

What Black-Scholes assumes that markets don’t deliver

• ▸Constant volatility — the biggest violation. Real volatility moves around constantly, and that is exactly what burns option buyers.
• ▸European exercise only — the textbook model assumes no early exercise. Most U.S. equity options are American and can be exercised early.
• ▸Log-normal returns — it assumes a tidy bell curve and misses fat tails. Real markets gap, crash, and jump more often than the model expects.
• ▸No transaction costs or taxes — spreads, commissions, and slippage are real and the model ignores all of them.
• ▸The volatility smile/skew is real: identical-expiry options at different strikes trade at different implied vols, which a single-σ model cannot capture.