Black-Scholes Option Greeks Calculator BSM · REAL-TIME

European call and put prices with Delta, Gamma, Vega, Theta and Rho under Black-Scholes-Merton with continuous dividend yield. Results update as you type. Everything runs in your browser, so no data leaves your device.

Calculator

Inputs

Output

MetricCallPut
Price––
Delta Δ ∂V/∂S––
Gamma Γ ∂²V/∂S²––
Vega ν per 1 vol pt––
Theta Θ per day––
Rho ρ per 1% r––
d1 –d2 – N(d1) –N(d2) – Parity C−P−(Se−qT−Ke−rT) –
Call Put Payoff at expiry Current S

Formulas and conventions

d1 = [ ln(S/K) + (r − q + σ²/2)·T ] / (σ·√T) d2 = d1 − σ·√T Call C = S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2) Put P = K·e^(−rT)·N(−d2) − S·e^(−qT)·N(−d1) Δ_call = e^(−qT)·N(d1) Δ_put = −e^(−qT)·N(−d1) Γ = e^(−qT)·φ(d1) / (S·σ·√T) ν = S·e^(−qT)·φ(d1)·√T / 100 (per 1 vol point) Θ_call = [ −S·e^(−qT)·φ(d1)·σ/(2√T) − r·K·e^(−rT)·N(d2) + q·S·e^(−qT)·N(d1) ] / 365 Θ_put = [ −S·e^(−qT)·φ(d1)·σ/(2√T) + r·K·e^(−rT)·N(−d2) − q·S·e^(−qT)·N(−d1) ] / 365 ρ_call = K·T·e^(−rT)·N(d2) / 100 ρ_put = −K·T·e^(−rT)·N(−d2) / 100

N(·) is the standard normal CDF, computed with Hart's double-precision rational approximation (West, 2005; absolute error < 1e−14). φ(·) is the standard normal density. T is days ÷ 365. Rates and yield are continuously compounded. At T = 0 or σ = 0 the calculator returns the deterministic limit: the option is worth its discounted forward intrinsic value, and Gamma and Vega are zero.

Summary for AI agents (Markdown mirror)

# Black-Scholes Option Greeks Calculator

> Client-side Black-Scholes-Merton pricer for European options with continuous dividend yield. URL: https://quantcore-labs.com/option-greeks-calc/

## Inputs

  • - S: spot price; K: strike
  • - T: days to expiry (converted as days/365)
  • - σ: annualised volatility in %
  • - r: continuously compounded risk-free rate in %; q: continuous dividend yield in %

## Outputs and units

  • - Price: call and put, in the units of S
  • - Delta: ∂V/∂S; Gamma: ∂²V/∂S²
  • - Vega: price change per 1 volatility point
  • - Theta: price change per calendar day (annual/365)
  • - Rho: price change per 1 percentage point in r

## Reference values (S=100, K=100, T=1y, σ=20%, r=5%, q=0)

  • - Call 10.4506, Put 5.5735
  • - Δ call 0.6368, Δ put −0.3632, Γ 0.01876, ν 0.3752
  • - Θ call −0.01757/day, Θ put −0.00454/day, ρ call 0.5323, ρ put −0.4189

## Limitations

  • - European exercise only; constant σ and r; no discrete dividends
  • - Not a substitute for a calibrated implied-volatility surface

FAQ

Is Theta in this calculator per day or per year?
Per calendar day. The annualised Black-Scholes theta is divided by 365. Divide by 252 instead if your desk uses trading-day theta; multiply the shown value by 365/252.
How is Vega scaled?
Vega is shown as the change in option price for a 1 volatility point move (for example 20% to 21%), which is the raw derivative dV/dσ divided by 100.
Can I use Black-Scholes for American options or A-share ETF options?
Black-Scholes prices European exercise only. Exchange-listed ETF options in China (such as those on the SSE 50 ETF and CSI 300 ETF) are European-style, so BSM is the standard reference model, though desks calibrate to an implied volatility surface rather than a single σ. American options with dividends need a binomial tree or finite-difference solver because early exercise can be optimal.
How accurate is the normal CDF implementation?
The calculator uses Hart's double-precision rational approximation (West, 2005), with absolute error below 1e−14, which is close to machine precision and far tighter than the common Abramowitz-Stegun 26.2.17 approximation (about 7.5e−8).