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Financial Mathematics by Roman N. Makarov, Giuseppe Campolieti

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Chapter 16

Alternative Models of Asset Price Dynamics

In the study of stochastic processes and their applications to finance, geometric Brownian motion (GBM) is the simplest model used for continuous-time asset pricing. The volatility in the GBM model is constant, i.e., the diffusion coefficient is a linear function of the underlying asset price, as is the drift coefficient. For a long period of time the GBM model has stood as one of the few known continuous-time stochastic models which admits exact analytically tractable transition probability density functions and closed-form pricing formulae for various standard, barrier, and lookback European-style options. However, despite its simplicity, it is commonly recognized that the GBM model only ...

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