This Black-Scholes calculator prices European call and put options from spot, strike, time, rate and volatility — with d1, d2 and the deltas that traders actually steer by.

Call price –
Put price –
d1 / d2 –
Delta (call / put) –

Model assumptions, stated plainly: European exercise, no dividends, constant volatility and rate, lognormal prices, frictionless markets. Reality files its objections (volatility smiles, early exercise, dividend dates) — the model remains the quoting language anyway, with implied volatility as its dialect.

The 1973 Formula That Ate Finance

Call = S·N(d1) − K·e^(−rT)·N(d2); the put follows by symmetry (and put-call parity audits both). Profits from plain share trades are our Stock Profit Calculator. Part of our Finance Calculators collection.

Worked Example (Verified — the Textbook Case)

  • S = K = 100, T = 1 yr, r = 5%, σ = 20% (defaults): d1 = 0.35, d2 = 0.15 → Call = $10.45, Put = $5.57, call delta 0.637 — the most reproduced numbers in derivatives courses, matching reference tables to the cent
  • Parity audit: Call − Put should equal S − K·e^(−rT) = 100 − 95.12 = 4.88; 10.45 − 5.57 = 4.88 ✔ — when a model's two outputs check each other, quote both with confidence
  • Volatility is the trade: raise σ to 30% and the call becomes ≈ $14.23 with spot unchanged — option buyers purchase movement, not direction alone

Model Literacy

  • Implied volatility inverts the formula: markets quote σ, the model just translates — the “vol smile” is the market editing Black-Scholes in public.
  • Delta ≈ the hedge ratio (0.637 → short ~64 shares per 100 calls) and a rough risk-neutral probability of finishing in the money (N(d2) is the cleaner version of that idea).
  • American options on dividend payers can be worth more exercised early — Black-Scholes prices the European contract; know which one you hold.
  • Theta, vega, gamma complete the Greeks: time decay, vol sensitivity, delta's own delta. Price is the headline; Greeks are the risk department.

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Frequently Asked Questions

How does the Black-Scholes formula work?

d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T), d2 = d1 − σ√T; Call = S·N(d1) − K·e^(−rT)·N(d2), and Put = K·e^(−rT)·N(−d2) − S·N(−d1). The defaults (at-the-money, 1 year, 5%, 20% vol) give the textbook Call $10.45 / Put $5.57. This calculator also returns delta.

What volatility should I enter?

For pricing consistently with the market, the option's implied volatility (quoted by brokers) — historical volatility is the fallback estimate. Since σ is the only unobservable input, it carries the model's whole argument with reality; small σ changes move prices materially (vega).

What does delta mean?

The option price's sensitivity to a $1 move in the underlying, and the hedge ratio: delta 0.637 means the call gains ≈ 64 cents per $1 rise, and a market-maker shorts ~64 shares per 100 calls to neutralize. Put deltas are negative (call delta − 1 here).

Does Black-Scholes work for American options?

It's exact (within its assumptions) for European options. American calls on non-dividend stocks are never optimally exercised early, so the price matches; American puts and dividend-payer calls can exceed the European value — binomial models handle those.

Why do options on the same stock show different implied vols?

The volatility smile/smirk: markets price crash protection and fat tails that the lognormal model denies, so far-out-of-the-money puts carry higher implied σ. It's the model's known fiction, quoted anyway because a shared fiction with a correction chart beats no language at all.

Is this Black-Scholes calculator free?

Yes — free, instant, no sign-up.