Understanding Geometric Brownian Motion (GBM) in Stock Price Analysis

Geometric Brownian Motion (GBM) is a stochastic process widely used in financial modeling, particularly in stock price analysis. The process is often employed to model stock prices due to its simplicity and tractability. GBM is characterized by the stochastic differential equation (SDE):

Understanding Geometric Brownian Motion

Geometric Brownian motion can be described by the following SDE:

[ dS_t mu S_t dt sigma S_t dW_t ]

where:

( S_t ) is the stock price at time ( t ). ( mu ) is the drift term, representing the expected return. ( sigma ) is the volatility, representing the standard deviation of returns. ( W_t ) is a standard Wiener process, or Brownian motion.

The solution to this SDE provides the log-returns of the stock price, which can be expressed as:

[ log(S_t) log(S_0) (mu - frac{sigma^2}{2})t sigma W_t ]

Data Collection and Preparation

To conduct the analysis, historical stock price data is required. This data should be collected over a period that adequately captures the underlying market dynamics. Daily prices are preferable as they better capture the continuous nature of the GBM process.

The first step is to calculate the log returns of the stock prices:

[ R_t logleft(frac{S_t}{S_{t-1}}right) ]

Statistical Tests for Normality

Under GBM, the log returns should follow a normal distribution. This assumption can be tested using the Shapiro-Wilk Test, Kolmogorov-Smirnov Test, Anderson-Darling Test, and visual inspection using Q-Q plots. These tests help ensure that the data adheres to the theoretical distribution of GBM.

Visual inspection of the Q-Q plot can provide insights into the normality of the data. If the points lie close to the 45-degree line, the data is considered normally distributed.

Check for Autocorrelation

Under GBM, the log returns should exhibit no autocorrelation. The Durbin-Watson statistic and Ljung-Box test can be used to test for the presence of autocorrelation. These tests help confirm that the returns are not dependent on their own previous values.

Parameter Estimation

The drift ( mu ) and volatility ( sigma ) can be estimated using the sample mean and sample standard deviation of the log returns, respectively.

Modeling and Simulation

Fitting a GBM model to the data involves estimating the parameters using the robust methods mentioned above. Once the parameters are estimated, the model can be used to simulate paths of the stock price. Comparing the empirical distribution of log returns with the theoretical distribution from the GBM model is crucial for validation.

Goodness of Fit Tests

To assess the goodness of fit of the model, goodness-of-fit tests like the Chi-squared test can be employed. These tests help compare the observed distribution of returns with the expected distribution from the GBM model, ensuring a high degree of accuracy.

Visual Analysis

A visual analysis of the log returns can provide additional insights. A random scatter around zero with no clear trends suggests that the GBM assumption is valid. Any observed patterns may indicate deviations from the GBM model.

Conclusion

If all the statistical tests confirm that the log returns are normally distributed, exhibit no autocorrelation, and fit the GBM model parameters well, it is reasonable to conclude that the stock prices follow a geometric Brownian motion. However, it is important to recognize that real-world financial data can be complex. Market inefficiencies, jumps, and other factors can still cause deviations from the GBM model.

By following this comprehensive guide, financial analysts and researchers can effectively test the GBM hypothesis and better understand stock price dynamics.