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  "body": "\n## Summary\n\nGeometric Brownian motion is a positive process whose logarithm is a Brownian motion with drift.\nIt solves the stochastic differential equation $dS_t = \\mu S_t \\, dt + \\sigma S_t \\, dW_t$.\nIt is the model of stock prices in the Black–Scholes theory of option pricing.\n\n## Explanations\n\n### L1 human\n\nIn geometric Brownian motion, the random changes are proportional to the current value, so it grows or shrinks by percentages rather than by fixed amounts.\nThis is why it never becomes negative, which suits prices of shares.\nOver a time $t$, the value is multiplied by a [[log-normal|4W40NHWPQT]] random factor.\nFor example, with drift $\\mu = 0.05$ and volatility $\\sigma$ the expected value grows like $e^{0.05 t}$, while the median value grows only like $e^{(0.05 - \\sigma^2/2) t}$.\n",
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