Black-Scholes Options Calculator
European call + put prices and the 5 Greeks (delta, gamma, vega, theta, rho) under the Black-Scholes-Merton model.
Result
- Call price$5.8836
- Put price$8.5475
- dโ-0.06033
- dโ-0.23711
- N(dโ)0.47595
- N(dโ)0.40629
- โ Greeks (call) โ
- Delta (call)ฮ option price per $1 move in S.0.47595
- Gammaฮ delta per $1 move in S (same for call & put).0.022527
- Vega (per 1% ฯ)ฮ option price per 1 vol-point increase.0.28158
- Theta (per day, call)Option-value decay per calendar day.-0.02443
- Rho (per 1% rate, call)ฮ option price per 1% rate increase.0.20855
- โ Greeks (put) โ
- Delta (put)-0.52405
- Theta (per day, put)-0.01177
- Rho (per 1% rate, put)-0.30476
- Put-call parity checkShould be ~0 (numerical noise).|$-2.6639 โ $-2.6639| = $0
Step-by-step
- dโ = [ln(S/K) + (r โ q + ฯยฒ/2)ยทT] / (ฯยทโT) = -0.06033.
- dโ = dโ โ ฯยทโT = -0.23711.
- Call = Sยทe^(โqยทT)ยทN(dโ) โ Kยทe^(โrยทT)ยทN(dโ) = $47.5946 โ $41.711 = $5.8836.
- Put = Kยทe^(โrยทT)ยทN(โdโ) โ Sยทe^(โqยทT)ยทN(โdโ) = $8.5475.
How to use this calculator
- Enter S = current underlying price, K = strike price.
- T = decimal years to expiry. 30 days = 30/365 = 0.0822.
- r = annualized risk-free rate (use the 3-mo T-bill yield, ~5% in 2026).
- ฯ = annualized volatility. Use historical (sample std dev ร โ252) or implied from option chain.
- q = continuous dividend yield. Use 0 for non-dividend stocks; ~1.5-2% for S&P 500.
About this calculator
The Black-Scholes-Merton model (Fischer Black, Myron Scholes, Robert Merton โ Nobel 1997 to Scholes & Merton; Black had died in 1995 and the prize is not awarded posthumously) prices a European-style option as the discounted risk-neutral expected payoff under a geometric-Brownian-motion stock-price process. The 5 standard Greeks measure first-order sensitivity to the inputs: ฮ to underlying price, ฮ to delta itself, ฮฝ to volatility, ฮ to time, ฯ to rate. The model assumes constant volatility (the most-violated assumption โ implied-vol smiles are exactly the market's correction), no early exercise, no transaction costs, and log-normally distributed prices. For US single-stock options (most are American-style), the European-style Black-Scholes price is a lower bound; the early-exercise premium is generally small for non-dividend-paying calls but can be material for puts and for calls on dividend-paying stocks just before ex-date.