{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "RJ9767FFTF"
    }
  ],
  "body": "\n## Summary\n\nThe Black–Scholes equation is the linear parabolic equation $V_t + \\frac{1}{2}\\sigma^2 S^2 V_{SS} + r S V_S - r V = 0$ for the price $V(S, t)$ of a financial option.\nIt assumes that the price $S$ of the underlying asset follows [[geometric Brownian motion|D48DR46B97]] with volatility $\\sigma$.\nIt also assumes that money earns the constant interest rate $r$.\nIt is solved backward in time from the payoff at expiry.\nA change of variables, starting with $x = \\log S$ and reversing time, turns it into the [[heat equation|VP2HN86VAG]].\n\n## Explanations\n\n### L1 human\n\nAn option is a contract whose value at a future date $T$ depends on the price of a stock then.\nThe equation says how much the option is worth today.\nThe key idea is that a portfolio of the option and the stock can be kept free of risk, so it must grow at the rate $r$.\nFor a call option with strike $K$, the final value is $\\max(S - K, 0)$, and the equation gives the Black–Scholes formula for earlier times.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "35Q91",
      "91G20"
    ],
    "name": "Black–Scholes equation",
    "same_as": {
      "wikidata": "Q17005676",
      "wikipedia": "Black–Scholes equation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_equation"
      }
    ],
    "tag": "ZFTNJ2YREG"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/ZFTNJ2YREG/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/ZFTNJ2YREG.md",
  "msc": [
    "35Q91",
    "91G20"
  ],
  "name": "Black–Scholes equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZFT/ZFTNJ2YREG.md",
  "status": null,
  "tag": "ZFTNJ2YREG",
  "uri": "m2://ZFTNJ2YREG",
  "url": "https://tamnd.github.io/math2/wiki/black-scholes-equation/",
  "writing": {
    "tag": "ZFTNJ2YREG",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}