This article introduces mixing theorems, a theoretical and computational approach to certain advanced option models. To begin, the Black-Scholes-Merton family of models is a well-known and sensible starting framework for understanding option prices. The framework relies on the assumption that the underlying stock price (or security price) follows a process known as geometric Brownian motion (GBM). This model has some very strong points in its favor: (i) it's consistent with stocks as limited liability securities and so the prices never fall below zero, (ii) it has uncorrelated returns, which have strong statistical support over many time scales, and (iii) it's very tractable computationally.
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