Geometric Brownian Motion is a stochastic process that models a randomly varying quantity following a Brownian motion with drift. It is a popular stochastic method for simulating stock prices that follow a trend while experiencing a random walk of up-and-downs characterizing risk.
The following notes were used for my implementation of GBM:
http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-BM.pdf
http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-GBM.pdf
GBM Simulation Results on SPY (S&P 500; Jun 9, 2023)
Short-Term Valuation Cycle
Using the simulation shown above, we can estimate the probability that the asset value would be greater than the current value during the N-day period following the current time.
At every time t, observe the asset value's historical data during the past 6-months (120 days). Through the GBM simulation, estimate the probability that the asset value would be greater than the current value during the 3-month (60-days) period following t. This probability is impacted by the short-term mean return and variation in return (or volatility and risk) according to the principles of GBM.
Repeat the same procedure for every t and we get the following output.
The graph on the top shows the probabilities estimated through the simulation each day for the S&P 500. Notice that those probabilities appear to be leading indicators; once the probabilities reach a significant value (below 0.30 and above 0.80), the stock index reverses direction!