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edited Jun 10, 2015 at 7:08

Maybe this is rather a comment then an answer but twothree points:

GBM is a stochastic model for stock prices. It is used to price derivatives in an arbitrage free setting. In this case you look at a process whose expected return is just the risk free rate (due to no-arbitrage). Forecast this price is trivial.
It is debatable how forecastable stock prices are. If you want to learn about forecasting in a statistical sense then you could look at Rob Hyndman's resources (papers, R packages, free online book).

one more point. You write: "As you see in the plot of the mean function, it says that the stock will only varies in a range of 0.05 cents in 17 days which is totally wrong taking into account that this stock varies more than 1 dollar in a normal trading". This is the dominance of variance over the mean that we often see in stock prices and for sure in stochastic models. During short time periods what you see in your (real or simulated) price path is mainly volatility (the sigma) while you "feel" the mu only for longer horizons. Just forecasting the price for the future with today's value will give you a forecast as precise as the mean that you apply - it is all noise on your timeframe.

Maybe this is rather a comment then an answer but two points:

GBM is a stochastic model for stock prices. It is used to price derivatives in an arbitrage free setting. In this case you look at a process whose expected return is just the risk free rate (due to no-arbitrage). Forecast this price is trivial.
It is debatable how forecastable stock prices are. If you want to learn about forecasting in a statistical sense then you could look at Rob Hyndman's resources (papers, R packages, free online book).

Maybe this is rather a comment then an answer but three points:

GBM is a stochastic model for stock prices. It is used to price derivatives in an arbitrage free setting. In this case you look at a process whose expected return is just the risk free rate (due to no-arbitrage). Forecast this price is trivial.
It is debatable how forecastable stock prices are. If you want to learn about forecasting in a statistical sense then you could look at Rob Hyndman's resources (papers, R packages, free online book).

one more point. You write: "As you see in the plot of the mean function, it says that the stock will only varies in a range of 0.05 cents in 17 days which is totally wrong taking into account that this stock varies more than 1 dollar in a normal trading". This is the dominance of variance over the mean that we often see in stock prices and for sure in stochastic models. During short time periods what you see in your (real or simulated) price path is mainly volatility (the sigma) while you "feel" the mu only for longer horizons. Just forecasting the price for the future with today's value will give you a forecast as precise as the mean that you apply - it is all noise on your timeframe.

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created Jun 10, 2015 at 6:44

Maybe this is rather a comment then an answer but two points:

GBM is a stochastic model for stock prices. It is used to price derivatives in an arbitrage free setting. In this case you look at a process whose expected return is just the risk free rate (due to no-arbitrage). Forecast this price is trivial.
It is debatable how forecastable stock prices are. If you want to learn about forecasting in a statistical sense then you could look at Rob Hyndman's resources (papers, R packages, free online book).