{
  "slug": "black-scholes",
  "name": "Black-Scholes Option Price",
  "category": "Derivatives",
  "description": "Closed-form European call/put pricing under the geometric Brownian motion assumption.",
  "formula": "C = S × N(d1) - K × e^(-rT) × N(d2)\nP = K × e^(-rT) × N(-d2) - S × N(-d1)",
  "formula_latex": "C = S\\,N(d_1) - K e^{-rT} N(d_2),\\quad d_1 = \\dfrac{\\ln(S/K) + (r + \\sigma^2/2)T}{\\sigma\\sqrt{T}},\\quad d_2 = d_1 - \\sigma\\sqrt{T}",
  "parameters": [
    {
      "name": "Spot price",
      "symbol": "S",
      "type": "number",
      "default": 100
    },
    {
      "name": "Strike",
      "symbol": "K",
      "type": "number",
      "default": 100
    },
    {
      "name": "Time to expiry (years)",
      "symbol": "T",
      "type": "number",
      "default": 0.25,
      "min": 0.0001
    },
    {
      "name": "Risk-free rate",
      "symbol": "r",
      "type": "percent",
      "default": 0.043
    },
    {
      "name": "Implied volatility",
      "symbol": "σ",
      "type": "percent",
      "default": 0.2,
      "min": 0.0001
    },
    {
      "name": "Option type",
      "symbol": "type",
      "type": "number",
      "description": "1 for call, 0 for put",
      "default": 1
    },
    {
      "name": "Continuous dividend yield",
      "symbol": "q",
      "type": "percent",
      "default": 0,
      "required": false
    }
  ],
  "outputs": [
    {
      "name": "Option price",
      "symbol": "Price",
      "type": "number"
    },
    {
      "name": "Delta",
      "symbol": "Δ",
      "type": "number"
    },
    {
      "name": "Gamma",
      "symbol": "Γ",
      "type": "number"
    },
    {
      "name": "Vega",
      "symbol": "ν",
      "type": "number"
    },
    {
      "name": "Theta",
      "symbol": "Θ",
      "type": "number"
    },
    {
      "name": "Rho",
      "symbol": "ρ",
      "type": "number"
    }
  ],
  "worked_example": {
    "narrative": "ATM 3-month call, S=K=100, r=4.3%, σ=20%, q=0. Greeks expressed in conventional per-1pt units (vega per 1 vol-pt, theta per day, rho per 1pp rate move).",
    "inputs": {
      "S": 100,
      "K": 100,
      "T": 0.25,
      "r": 0.043,
      "sigma": 0.2,
      "type": 1,
      "q": 0
    },
    "expected_outputs": {
      "price": 4.52,
      "delta": 0.5625,
      "gamma": 0.0394,
      "vega": 0.1971,
      "theta": -0.0341,
      "rho": 0.1242
    }
  },
  "assumptions": [
    "Constant volatility",
    "No dividends (or constant continuous yield)",
    "Continuous, frictionless trading",
    "Lognormal terminal stock price",
    "Constant risk-free rate"
  ],
  "limitations": [
    "Volatility smile/skew shows real markets violate constant-vol assumption",
    "European-only (no early exercise)"
  ],
  "alternatives": [
    {
      "slug": "black-76",
      "when_to_prefer": "Options on futures"
    },
    {
      "slug": "binomial-tree",
      "when_to_prefer": "American-style options or path-dependent payoffs"
    }
  ],
  "related_terms": [
    "black-scholes",
    "implied-volatility",
    "delta-hedging",
    "greeks",
    "european-option"
  ],
  "references": [
    {
      "title": "The Pricing of Options and Corporate Liabilities",
      "author": "Black & Scholes",
      "year": 1973
    },
    {
      "title": "Theory of Rational Option Pricing",
      "author": "Robert C. Merton",
      "year": 1973
    }
  ]
}