Predicting Stock Market Movements with a Markov‑Chain State‑Transition Model

Markov Chain Basics

A Markov chain is a stochastic system where the next state depends only on the current state, not on the sequence of previous states. The article illustrates this with a weather example: if today is sunny, tomorrow has an 80% chance of remaining sunny and a 20% chance of rain; if today is rainy, tomorrow has a 30% chance of becoming sunny and a 70% chance of staying rainy.

Transition Matrix

The probability relationships are expressed in a transition matrix where each row represents the current state and each column the possible next state. The matrix entries are the transition probabilities.

Multi‑Step Prediction

Multiplying the transition matrix by itself yields the two‑step transition probabilities. For the weather example, the probability of sunny weather two days later is 0.70 (0.8×0.8 + 0.2×0.3), and rain is 0.30.

Steady‑State Convergence

Raising the matrix to higher powers shows convergence to a steady‑state distribution where probabilities no longer change. After five and ten iterations the matrix approaches a stable distribution (≈60% sunny, 40% rainy in the example), demonstrating that the chain “forgets” its initial state.

Applying the Model to the Stock Market

To use a Markov chain for market analysis, the first step is to define discrete states for daily price movement:

Big Drop (↓↓)

Small Drop (↓)

Neutral (→)

Small Rise (↑)

Big Rise (↑↑)

Each trading day is assigned to one of these states, and a transition matrix is built from historical sequences.

Python Implementation – Step‑by‑Step

Step 1: Environment Setup

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns

Step 2: State Creation

def create_states(prices, n_states=5):
    """Convert continuous price changes into discrete states"""
    percent_change = prices.pct_change().dropna()
    try:
        if len(np.unique(percent_change.values)) < n_states:
            states = pd.cut(percent_change.values, bins=n_states, labels=range(n_states), duplicates='drop')
        else:
            states = pd.qcut(percent_change.values, q=n_states, labels=range(n_states), duplicates='drop')
    except ValueError:
        states = pd.cut(percent_change.values, bins=n_states, labels=range(n_states), duplicates='drop')
    return pd.Series(states, index=percent_change.index).astype(int)

Step 3: Transition Matrix Construction

def build_transition_matrix(states, n_states):
    if states.empty or len(states) < 2:
        return np.full((n_states, n_states), np.nan)
    matrix = np.zeros((n_states, n_states))
    for i, j in zip(states[:-1], states[1:]):
        if pd.notna(i) and pd.notna(j):
            matrix[int(i), int(j)] += 1
    row_sums = matrix.sum(axis=1, keepdims=True)
    return np.divide(matrix, row_sums, out=np.zeros_like(matrix), where=row_sums != 0)

Step 4: Model Integration

# Load historical price data (example ticker)
prices = load_data_from_source("TICKER_SYMBOL")['Close']
state_labels = ["Big Drop", "Small Drop", "Neutral", "Small Rise", "Big Rise"]
states_series = create_states(prices, n_states=len(state_labels))
trans_matrix = build_transition_matrix(states_series, n_states=len(state_labels))
current_state_idx = states_series.iloc[-1]
print(f"Current State: {state_labels[current_state_idx]}")
print("
Next State Probabilities:")
for i, label in enumerate(state_labels):
    print(f"- {label}: {trans_matrix[current_state_idx, i]:.2%}")

Step 5: Visualization

def plot_transition_matrix(matrix, labels):
    """Create a heatmap of the transition matrix"""
    plt.figure(figsize=(10, 8))
    sns.heatmap(matrix, annot=True, cmap="YlGnBu", fmt=".2f", xticklabels=labels, yticklabels=labels, vmin=0, vmax=1)
    plt.title("Transition Probability Matrix")
    plt.xlabel("Next State")
    plt.ylabel("Current State")
    plt.tight_layout()
    plt.show()

Step 6: Long‑Term Steady‑State Calculation

def compute_steady_state(trans_matrix, iterations=20):
    """Iteratively raise the matrix to obtain the steady‑state distribution"""
    current = trans_matrix.copy()
    for _ in range(iterations):
        current = np.dot(current, trans_matrix)
    return current[0]
steady_state = compute_steady_state(trans_matrix)
print("Long‑term Market State Distribution:")
for state, prob in zip(state_labels, steady_state):
    print(f"- {state}: {prob:.2%}")

Empirical Case Study

Using recent market data (as of 2025‑05‑06) the model classified the current day as a “Big Drop” with a closing price of $9.18 and a –1.40% change. The transition matrix for the next day from this state is:

Big Drop    0.1940  0.1940  0.1679  0.1530  0.2910
Small Drop  0.1940  0.1716  0.1903  0.2201  0.2239
Neutral     0.1679  0.2015  0.2052  0.2537  0.1716
Small Rise  0.2201  0.2164  0.2201  0.2276  0.1157
Big Rise    0.2230  0.2156  0.2156  0.1450  0.2007

From this row, the highest probability (29.10%) is a “Big Rise” the next day, while the combined probability of further declines (Big Drop + Small Drop) is about 39%.

Model Evaluation

Risk Assessment – Quantifies the chance of extreme moves from the current state.

Pattern Recognition – Checks whether observed transitions match historical probabilities.

Strategy Formulation – Enables building trading systems that consider state‑dependent dynamics.

Limitations

The market evolves; transition probabilities may change over time.

External shocks (e.g., pandemics) can invalidate historical patterns.

Choice of state granularity and time horizon heavily influences results.

The memoryless assumption oversimplifies complex market dynamics.

Conclusion

The Markov‑chain state‑transition model offers a probabilistic lens for short‑term market movements and long‑term equilibrium analysis. While it cannot deliver deterministic price forecasts, it provides valuable risk‑aware insights that can complement other analytical tools. Future work may combine Markov chains with advanced machine‑learning techniques to capture richer market dynamics.