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  "body": "\n## Definition\n\nLet $\\sigma \u003e 0$ and $r \\ge 0$ be constants and let $T \u003e 0$.\nThe *Black–Scholes equation* is the [[parabolic equation|1CCTH8X2FT]] for a function $V(S, t)$ with $S \u003e 0$ and $0 \\le t \u003c T$:\n$$\\frac{\\partial V}{\\partial t} + \\frac{1}{2}\\sigma^2 S^2 \\frac{\\partial^2 V}{\\partial S^2} + r S \\frac{\\partial V}{\\partial S} - r V = 0.$$\nHere $S$ is the price of the underlying asset, modelled by [[geometric Brownian motion|D48DR46B97]] with volatility $\\sigma$.\nThe constant $r$ is the risk-free interest rate, and $V$ is the price of a derivative security.\nIt is solved with a terminal condition $V(S, T) = \\Phi(S)$ given by the payoff $\\Phi$.\n\n## Remarks\n\nExample: $V(S, t) = S$ is a solution: $V_t = 0$, $V_{SS} = 0$, and $r S \\cdot 1 - r S = 0$.\nIt is the value of holding one share.\nAnother solution is $V(S, t) = K e^{-r(T - t)}$, the value of a bond paying $K$ at time $T$: $V_t - r V = 0$.\nNon-example: $V(S, t) = S^2$ fails, since $\\frac{1}{2}\\sigma^2 S^2 \\cdot 2 + r S \\cdot 2S - r S^2 = (\\sigma^2 + r) S^2 \\ne 0$.\nWith $x = \\log S$ and $\\tau = T - t$, the equation becomes a heat equation with constant drift and decay terms, so it is backward parabolic in $t$.\n",
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