05. Calibrating GBM to data

Calibration of Geometric Brownian Motion to Data

Understanding Momentum-Based Trading with Geometric Brownian Motion

Learn how to calibrate Geometric Brownian Motion (GBM) to data for momentum-based trading. GBM is a model for predicting stock prices by considering both deterministic and stochastic elements.

Key Concepts:

• Geometric Brownian Motion (GBM): A mathematical model used for simulating stock prices, considering the drift (momentum) and volatility.
• Stochastic Differential Equation (SDE): Describes the change in stock prices, incorporating both drift (mu) and volatility (sigma).
• Calibration: Adjusting GBM parameters to fit historical stock data, such as the S&P 500 index.

Steps in Calibration:

1. Import Libraries:

   • Numpy: For numerical calculations.
   • Scipy: For statistical functions, especially normal distribution.
   • CSV and Contextlib: For reading and handling data files.

2. Define GBM Class:

   • Initialize mu and sigma.
   • Create random number generator for simulation.

3. Calibration Function:

   • Calculate mean and standard deviation of the log of increments.
   • Derive mu and sigma from these statistics.

4. Load Data:

   • Read historical S&P 500 index values from a CSV file.
   • Use GBM class to calibrate these data points.

5. Analysis:

   • Estimated momentum is around 2.2%.
   • Estimated volatility is about 6.7%, indicating market risk.

Upcoming Topics:

• Discuss options for risk hedging.
• Explore qualitative risk management strategies.

This learning material provides insights into coding applications for finance. Happy coding!

QUESTION:

Download the daily closing prices of Apple in 2024 from e.g. Yahoo finance. Is GBM a good model for the prices? If it is, calibrate a GBM to the prices.