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I would recommend you the following econometrics textbook Basics Econometrics, with a particular focus on multinomial logit / probit models. I guess the challenging part in your case will consist of specifying the exogenous variables, collecting data, before doing the computations. The latter being quick to perform. As far as I am concerned it's better to ...

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In fact there is at least one application, namely in the pricing of reversions. The simplest case of a reversion is where there is no ground rent, and where exclusive possession (including the right to sell the property, or to rent it out) is deferred to some future date. Practically, this means foregoing any income from the property until the reversion ...

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Fundamentals usually do not enter the parameters of GBM. But it depends on the purpose: if you want to price options, then the drift is the risk-less rate and volatility is implied from other traded derivatives. if you want to use GBM for risk management then you usually apply statistical methods for $\sigma^2$.

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It turns out that GBM with constant drift and constant volatility is not really used in real life. It is well known that volatility as well as drift may vary over time. Hence, if you want to use a model with time-varying parameters, you need to come up with a model to define $\mu_t$ and $\sigma_t$. There are classic models that use some mean-reverting ...

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It does not necessarily have to be using historical data (you could use implied volatilities for example), but indeed fundamental analysis is not taken into account in geometric Brownian motions: you just assume returns are normally distributed with some mean and volatility and it does not change in time. So if you want to "incorporate" fundamental analysis ...

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