04. Confidence interval of the GBM price forecast

Understanding Stock Price Predictions with Geometric Brownian Motion

The geometric Brownian motion (GBM) model aids in predicting stock prices. Central to this model is the stock price formula:

• S(t): Future stock price.
• S(0): Current stock price.

Key Concepts

• Expectation: Calculated as the current price multiplied by the exponential of Mu*t, where Mu acts like a momentum or interest rate.
• Fluctuations: Price could deviate widely from the expected $100, perhaps falling between $50 and $150, or staying narrow between $99 and $101.

Confidence Interval

• Predictive Band: A 95% confidence interval outlines where prices can fluctuate, with a 2.5% probability to surpass lower or upper bounds.
• Symmetry or Customization: The choice of the symmetrical band (e.g., 2.5% on both sides) can be tailored based on requirements.

Calculation

• Lower Bound (L): Defined using the expectation, randomness of W(t), and the standard normal variable Z.
• Numerical Value: Calculated using Python's Scipy package with the PPF function, resulting in a typical bound of approximately 1.96 for statistical modeling.

Next, explore obtaining the values of Mu and Sigma to apply the model effectively.