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My immediate thoughts were to think of the limitations of a Brownian motion which is that it is continuous so that it doesn't take into account jumps in stock markets, obviously it is also not stationary which arises problems for forecasting and indeed it is symmetric which is not accurate of most data!

In light of those drawbacks, it is still clear to see that a geometric Brownian motion of the form

$$ S_t = S_0e^{X_t}$$ where $X_t$ is a Brownian motion with drift is still non-stationary and continuous. I assume that is symmetric (Im unsure really how this is determined), so with all of these factors in mind is it really any better?

Thanks in advance, I have only just read upon this process so forgive me if i have interpreted anything incorrectly.

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    I think the main reason why GBM is studied in financial context is that the process $S$ is non-negative and that European Call option prices can be determined explicitly (Black Scholes formula). – 2017-01-05
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    Ask [N](http://math.stackexchange.com/users/88078/nero)er[o](https://www.youtube.com/watch?v=R-Km9qo5JyA) – 2017-01-07

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The simpler, the better.

Mathematics are used as a tool , traders use it to have a starting point , and comfort their intuition. At the end of the day , it is all about gambling.

In the manner of most industries, you start simple and try to improve it. First of all, we want stock prices to be positive, hence a non-negative process is required. However, more importantly, a good model must be consistent with very liquid markets , mainly the option market.

At that stage , if we look at the possibilities, the one that comes first is normal distribution, it can be negative... the second one is lognormal.

The lognormal process is non-negative, high variance for high prices, low variance for low prices, it gives closed-form formulas for options prices that can be implemented in any old pocket calculators.

Even better, prior to the end of the 80s, the market option was flat, in other words, if I take any market option price during that period, and back out the lognormal volatility from it , this number is almost constant disregarding the strike of the option ! Why would one look for a complicated distribution ? I can have a bunch of options and just monitor one parameter.

Since the 1987 crisis, we have been observing a smile or skew in the market (Lognormal volatiliy wrt option strike) , and henceforth, this model is totally bogus.The lognormal model is only used as a way of quotation ,or a way to gauge the market volatility level, it became an indicator.

Finance covers also the rate market, and nobody utilizes the lognormal model( rates can be negative) but rather the normal model that is more representative of the rate option market.

Even if they used it, it would be for historical reasons, or some habits( bad ones ). Again, trading is an "art" (bad art exists?) .

As for the non-stationary feature, traders care more about consistency than the other features.

Indeed, one needs models to price and monitor a portfolio that may also contain illiquid products. These products can resemble simpler products that are quoted actively to the market. If one model cannot be consistent with the liquid market, there is no point pricing something more complex.

Having a model that is practical, consistent, stationary is impossible, and practitioners would rather choose a practical and consistent models.

Nowadays models are definitely not normal or lognormal models, but better versions of it that can include jumps, stochastic volatilities. It will not go further for practical reasons.

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I'd say non-negativity and time-varying (non-zero) expected value are two of the bigger reasons. It is still simple enough to allow for explicit representation of the pdf, which is useful.