The Black-Scholes-Merton model (1973) formulates European options valuation assuming a log-normal asset price path, constant risk-free rate, and continuous hedging dynamics.
Closed-Form Equations
Call (C) = S0 × N(d1) - K × e^(-rT) × N(d2)
Put (P) = K × e^(-rT) × N(-d2) - S0 × N(-d1)
d1 = [ ln(S0 / K) + (r + σ² / 2)T ] / ( σ √T )
d2 = d1 - σ √T
The Option Greeks Reference Matrix
| Greek Symbol | Mathematical Meaning | Primary Risk Factor | Trader Hedging Impact |
|---|---|---|---|
| Delta (Δ) | ∂V / ∂S | Directional Price Velocity | Hedge ratio for underlying shares |
| Gamma (Γ) | ∂²V / ∂S² | Rate of Delta Change | Acceleration risk on expiry week |
| Theta (Θ) | ∂V / ∂t | Calendar Time Decay | Daily dollar erosion of long options |
| Vega (ν) | ∂V / ∂σ | Implied Volatility Shift | Exposure to earnings crush and IV expansion |
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Interactive Model
Verify option pricing inputs and plot Greeks payoff curves with the Options Strategy Payoff Calculator.