Black-Scholes calculator
Black-Scholes option pricing calculator: European call and put prices with d1 and d2 plus Delta, Gamma, Vega, Theta and Rho, with a continuous dividend yield and a put-call parity check.
Runs in your browserEvery computation happens in your browser — your data never leaves this device.
Option parameters
The spot and strike prices are entered in yuan and converted to integer cents internally; rates and volatility are percentages. The normal distribution is implemented from scratch (Abramowitz & Stegun 7.1.26) with an absolute error of about 1e-7.
Price and Greeks
0.35000.150010.455.5710.450.63680.01880.3752-0.0176-6.41400.53230.0010.450.00NoWhat this tool does
- Check a listed option against its theoretical value to see whether the market is charging too much for time value.
- Work out the Delta and Gamma of a position to decide how much of the underlying to hedge.
- Use Vega and Theta to see whether a volatility trade earns from vol or gets eaten by time decay.
- Verify textbook exercises: with S=100, K=100, r=5%, σ=20% and T=1 the call should be 10.4506.
Example
Input
Spot 100, strike 100, annual volatility 20%, risk-free rate 5%, one year to expiry, no dividend, call
Output
d₁ 0.3500, d₂ 0.1500, call 10.45, put 5.57, Delta 0.6368, Gamma 0.0188, Vega (per 1%) 0.3752, Theta (per day) -0.0176, Rho (per 1%) 0.5323
The parity check reads 0.00: 10.45 − 5.57 equals 100 − 100×e^(−0.05), so the two prices are internally consistent.
Frequently asked questions
Is the normal distribution implemented from scratch, and is it accurate enough?
Yes: Φ(x) = 0.5×(1+erf(x/√2)) with erf from the five-term approximation of Abramowitz & Stegun, Handbook of Mathematical Functions, formula 7.1.26, whose stated absolute error is at most 1.5e-7. For option prices quoted to the cent that is far more precision than the display needs.
Why do Vega, Theta and Rho each appear in two forms?
Mathematically Vega is the price change for a 1.00 move in volatility — a hundred percentage points — which is unwieldy, so the tool also shows the change per one percentage point. Theta and Rho follow the same logic: Theta is divided by 365 for the daily decay and Rho by 100 for the change per one percentage point of rates.
Why does my broker show a different price?
Black-Scholes assumes constant volatility, continuous hedging and frictionless markets. Real markets have a volatility smile (each strike has its own implied volatility), bid-ask spreads, jump risk and, for American options, early exercise, so exchange prices usually deviate. This tool gives the theoretical price for the volatility you enter, not a forecast.
What goes in the dividend yield field?
The continuous dividend yield, used in Merton’s dividend-adjusted version: dividends make calls cheaper and puts more expensive because an option holder does not receive them. For a single stock an annualised yield is a reasonable approximation, and for a non-payer just leave it at 0.
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