Inputs
Results
| Call Price (C) | 9.226994 |
|---|---|
| Put Price (P) | 6.330069 |
| d₁ | 0.250000 |
| d₂ | 0.050000 |
Price vs Spot (S)
Chart shows model prices of call and put for a range of S using the parameters above.
Example Data Table
| Label | Spot S | Call Price | Put Price |
|---|---|---|---|
| 50% S | 50.000000 | 0.001628 | 46.114637 |
| 75% S | 75.000000 | 0.777716 | 22.385758 |
| 100% S | 100.000000 | 9.226994 | 6.330069 |
| 125% S | 125.000000 | 28.455155 | 1.053263 |
| 150% S | 150.000000 | 52.029628 | 0.122769 |
Formula Used
The Black–Scholes model prices European options (continuous dividend yield q):
C = S e^{-qT} N(d₁) − K e^{-rT} N(d₂)
P = K e^{-rT} N(−d₂) − S e^{-qT} N(−d₁)
d₁ = [ln(S/K) + (r − q + ½ σ²)T] / (σ√T)
d₂ = d₁ − σ√T
N(x) = cumulative standard normal distribution
r = risk‑free rate, q = dividend yield, σ = volatility, T = time in years
S = spot price, K = strike price
Assumptions include lognormal returns, constant volatility and rates, and European exercise.
How to Use
- Enter spot price, strike, risk‑free rate, dividend yield, volatility, and time to maturity in years.
- Press Calculate to compute call and put prices and view d₁ and d₂.
- Review the example table and chart to understand sensitivity to the underlying price.
- Export results or the table via CSV or PDF for reporting.
FAQs
No. The model prices European options only. Early exercise features require other models or approximations.
Enter annualized percentages (e.g., 5 for 5%). Time T should be in years (e.g., 0.5 for six months).
The calculator assumes a continuous dividend yield q. For discrete dividends, additional adjustments are needed.
The formula is undefined in those edge cases. Ensure both are positive to compute prices.
Markets use implied volatility curves, discrete dividends, transaction costs, and other adjustments not captured in the basic model.
This page focuses on pricing. You can extend it to compute Greeks (Delta, Gamma, Vega, Theta, Rho) using standard formulas.