The Black-Scholes-Merton Formula
The 1973 Black-Scholes-Merton formula (BSM) prices a European option on a non-dividend stock with closed-form. Merton's 1973 extension generalises it to any asset with a constant cost-of-carry b:
Call = S·e(b−r)T·N(d₁) − X·e−rT·N(d₂)
Put = X·e−rT·N(−d₂) − S·e(b−r)T·N(−d₁)
d₁ = [ln(S/X) + (b + σ²/2)T] / (σ√T), d₂ = d₁ − σ√T
Where S = spot, X = strike, T = time to expiry (years), σ = annualised volatility, r = risk-free rate, and b = cost of carry. For stocks with continuous dividend yield q: b = r − q. For stocks without dividends: b = r.
European vs American Options
European options can only be exercised at expiry. American options can be exercised at any time before expiry. For non-dividend calls, early exercise is never optimal, so European and American call prices are identical. For puts (and for dividend-paying calls), early exercise can be optimal — and American values exceed European.
This calculator prices American options with a 100-step Cox-Ross-Rubinstein binomial tree, the standard textbook method. Increase Binomial Steps for higher precision (500 steps gives ~0.01 accuracy for typical inputs).
The Greeks
Delta (Δ) — sensitivity of option price to a $1 change in spot. Ranges (0, 1) for calls, (−1, 0) for puts. At-the-money calls have Δ ≈ 0.5.
Gamma (Γ) — rate of change of Delta per $1 move in spot. Highest at the money, near expiry. High-gamma positions require frequent re-hedging.
Vega — sensitivity to a 1% change in volatility. Long options are long vega; short options (spreads, covered calls) are short vega. Expressed here as the price change per 1% absolute vol move (not per 1% relative).
Theta (Θ) — time decay per calendar day. Negative for long options — they bleed value as expiry approaches. Accelerates in the final weeks for ATM strikes.
Rho (ρ) — sensitivity to a 1% change in the risk-free rate. Small for short-dated equity options but meaningful for long-dated (LEAPS, ESOs) and FX options.
Advanced Greeks
Vanna (dΔ/dσ) — cross sensitivity. Important for hedging volatility skew: as vol rises, Delta changes even if spot is flat.
Charm (dΔ/dT) — delta decay over time. Used by dealers who want to know how their delta will drift by tomorrow's close without any market move.
Vomma (dVega/dσ) — convexity of vega. A barometer for the vol-of-vol exposure of a position. Strangles have high vomma; synthetic underlyings have zero.
Speed (dΓ/dS) — third-order sensitivity. Relevant for large spot moves where Gamma itself changes.
Zomma (dΓ/dσ) — sensitivity of Gamma to volatility changes. Used by exotic-desk risk managers.
Implied Volatility
Given an observed option premium in the market, implied volatility (IV) is the σ that makes BSM output equal the market price. There is no closed-form — this calculator uses bisection to converge to 8 decimal places within ~30 iterations. Options are quoted by IV because it normalises across strikes, maturities, and moneyness.
For index options, IV is typically in the 10–30% range. For single stocks, 20–60%. For earnings events or meme stocks, IVs of 100%+ are routine.
Worked Example — Apple (AAPL) Earnings Week
Suppose AAPL is trading at $175, you're looking at a $180 call expiring in 10 days, risk-free rate 4.5%, dividend yield 0.5%. Market IV is 45%.
- Option value ≈ $0.71
- Delta ≈ 0.23 (OTM call — low exposure)
- Gamma ≈ 0.09 (high — every $1 up in AAPL adds ~0.09 to delta)
- Theta ≈ −$0.10/day (earnings-week bleeding)
- Vega ≈ $0.10 per 1% vol change
After earnings (IV crush from 45% → 25%), even with AAPL flat, the option loses 20 × $0.10 = $2.00 in value — crushing the long call despite being "right" on direction. This is why earnings IV crush is devastating to premium buyers.
Model Limitations
- Assumes constant volatility — markets exhibit volatility smiles/skews, especially for puts.
- Assumes lognormal returns — real returns have fat tails (Taleb).
- Assumes continuous hedging — transaction costs and gaps break replication.
- No credit risk — for OTC options between counterparties, CVA/DVA apply.
For advanced use cases (stochastic vol, jumps, barriers), see Heston, SABR, or Monte Carlo methods. For portfolio-level options risk, see VaRisk Kancil.
Related Tools
- FX Options Calculator — Garman-Kohlhagen for currency options
- Bond YTM Calculator — fixed-income analytics
- Multi-Asset Return & Volatility — historical volatility input for σ
- Strategy Backtester — test option-like strategies on spot markets