Extensions of GBM
In an attempt to make GBM more realistic as a model for stock prices, one can drop the assumption that the volatility is constant. If we assume that the volatility is a deterministic function of the stock price and time, this is called a local volatility model. If instead we assume that the volatility has a randomness of its own—often described by a different equation driven by a different Brownian Motion—the model is called a stochastic volatility model.
Read more about this topic: Geometric Brownian Motion
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