Geometric Brownian motion

Quantitative Finance Glossary

Stochastic process for stock prices: dS_t = μ S_t,dt + σ S_t,dW_t, with closed-form solution S_t = S_0 exp!left[(μ - tfrac12σ^2)t + σ W_tright] — log-prices are Brownian motion with drift. Implies log-normal S_t, non-negative prices, and constant proportional volatility — the last assumption being conspicuously violated by the observed implied-vol skew. The Itô-to-Stratonovich drift correction -tfrac12σ^2 is the textbook 'GBM drift-trap' for new quants.

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