This project explores the use of Markov Chains to model and simulate the daily return of Apple's stock based on historical data. In the ever-changing financial market, stock price movements are influenced by company performance, economic indicators, and investor behavior. By leveraging a stochastic modeling approach, this analysis aims to uncover transition dynamics in Apple's returns, offering insights into trend behavior and helping investors make data-informed decisions.
The goal of this project is to:
- Analyze Apple's historical stock prices.
- Use Markov Chains to define states based on daily returns.
- Calculate transition probabilities and long-run behavior.
- Simulate possible future stock prices to guide investment decisions.
- Data Collection: Apple stock closing prices were retrieved from Google Finance for a period of ~2035 trading days.
- Daily Returns: Daily returns were calculated and visualized, revealing a fat-tailed, non-normal distribution.
- State Definition: Returns were divided into 4 states based on percentiles:
| State | Description | Percentile Range | Return Range (%) |
|---|---|---|---|
| 1 | Bear | 0–25% | -12.87 to -0.76 |
| 2 | Stagnant | 26–50% | -0.77 to 0.08 |
| 3 | Low Return | 51–75% | 0.09 to 1.03 |
| 4 | Bull | 76–100% | 1.04 to 11.97 |
- Transition Matrix (P-Matrix):
- Represents one-step transition probabilities between states.
- Ensures ergodicity and irreducibility.
| From \ To | State 1 | State 2 | State 3 | State 4 |
|---|---|---|---|---|
| State 1 | 0.297 | 0.169 | 0.220 | 0.313 |
| State 2 | 0.234 | 0.263 | 0.285 | 0.218 |
| State 3 | 0.207 | 0.308 | 0.266 | 0.219 |
| State 4 | 0.259 | 0.261 | 0.228 | 0.251 |
-
Steady-State Probabilities:
- Converge quickly to: π = (0.2493, 0.2504, 0.25, 0.2503)
- Mean recurrence times: all ~4 transitions.
-
Simulation:
- Starting stock price: $166, initial state: Bear (1)
- Simulated stock prices over 50 days using Monte Carlo simulation.
-
Single 50-Day Simulation: Showed both profit and loss opportunities.
-
1000 Simulations:
- Mean final price ≈ $161.94
- 5th percentile ≈ $79.9, 95th percentile ≈ $275.7
- Possible closing prices ranged from $37.25 to $462.95
- Markov Chains effectively modeled and simulated stock return dynamics.
- The system reached a steady-state after only a few transitions.
- While simulations give an insight into possible future behaviors, external factors like news, company policies, product launches, and geopolitical events must also be considered before making investment decisions.
- This methodology can be adapted for shorter durations or other stocks to assist in identifying profitable investment windows.
- Python (NumPy, Pandas, Matplotlib)
- Google Finance (for historical data)
- Markov Chain & Monte Carlo Simulation principles
- Apple Inc. stock data: Google Finance
- Markov Chain Theory: Applied Stochastic Processes (IE 5309)
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