{
  "slug": "implied-volatility",
  "name": "Implied Volatility (Newton-Raphson)",
  "category": "Derivatives",
  "description": "Inverts the Black-Scholes price to find the volatility consistent with a market option price.",
  "formula": "Find σ such that BS(S, K, T, r, σ) = MarketPrice",
  "formula_latex": "\\sigma : \\text{BS}(S, K, T, r, \\sigma) = \\text{Market Price}",
  "parameters": [
    {
      "name": "Market option price",
      "symbol": "P_mkt",
      "type": "number",
      "default": 4.61
    },
    {
      "name": "Spot",
      "symbol": "S",
      "type": "number",
      "default": 100
    },
    {
      "name": "Strike",
      "symbol": "K",
      "type": "number",
      "default": 100
    },
    {
      "name": "Time",
      "symbol": "T",
      "type": "number",
      "default": 0.25
    },
    {
      "name": "Risk-free rate",
      "symbol": "r",
      "type": "percent",
      "default": 0.043
    },
    {
      "name": "Type (1=call, 0=put)",
      "symbol": "type",
      "type": "integer",
      "default": 1
    }
  ],
  "outputs": [
    {
      "name": "Implied vol",
      "symbol": "IV",
      "type": "percent"
    }
  ],
  "worked_example": {
    "inputs": {
      "P_mkt": 4.61,
      "S": 100,
      "K": 100,
      "T": 0.25,
      "r": 0.043,
      "type": 1
    },
    "expected_outputs": {
      "iv": 0.2002
    }
  },
  "related_terms": [
    "implied-volatility",
    "vega",
    "vol-surface",
    "black-scholes"
  ]
}