Hidden Markov Model - Market Regime Trading

Decoding financial markets with probabilistic state machines

Multivariate Hidden Markov Model for market regime detection and systematic trading.

The Problem: Markets Don’t Move Randomly

Financial markets don’t just fluctuate—they exhibit regime-switching behavior. Bull markets surge with steady confidence and low volatility. Bear markets collapse with fear-driven spikes and violent drawdowns. The challenge? These regimes are invisible. We cannot simply observe whether we’re in a “bull” or “bear” state. All we see are daily returns, prices, and volatility measures. The regime remains hidden.

This is where I turned to Hidden Markov Models (HMMs)—a probabilistic framework that treats market regimes as latent states we must infer from observable data. I built this system to decode the unobservable, to predict not just what the market is doing, but what state it’s in.

Beyond Simple Returns: My Multivariate Approach

The conventional approach uses returns alone—a single feature fed into a univariate Gaussian HMM. It works, but it’s leaving information on the table. I extended this framework by incorporating wavelet energy as a second feature, creating a multivariate observation model:

\[\mathbf{o}_t = \begin{bmatrix} r_t \\ E_t \end{bmatrix}\]

Where:

  • \(r_t\) is the adjusted daily return
  • \(E_t\) is the wavelet energy capturing frequency-domain activity

This dual-feature approach allows my HMM to learn the covariance structure between returns and market activity. Bull markets? Positive returns with low energy (calm uptrends). Bear markets? Negative returns with high energy (volatile declines).

The Mathematical Engine

At the core of my implementation lies the complete HMM machinery. I didn’t just use a library—I understood and implemented the fundamental algorithms:

The Three Problems I Solved

Problem What I Computed Algorithm
Evaluation \(P(O \mid \lambda)\) — Likelihood of observations Forward Algorithm
Decoding \(Q^* = \arg\max P(Q \mid O, \lambda)\) — Most likely state sequence Viterbi Algorithm
Learning \(\lambda^* = \arg\max P(O \mid \lambda)\) — Optimal model parameters Baum-Welch (EM)

The Transition Matrix

My model learns a 2×2 transition matrix that captures regime persistence:

\[A = \begin{bmatrix} P(\text{BULL} \to \text{BULL}) & P(\text{BULL} \to \text{BEAR}) \\ P(\text{BEAR} \to \text{BULL}) & P(\text{BEAR} \to \text{BEAR}) \end{bmatrix}\]

The diagonal elements reveal how “sticky” each regime is—markets don’t flip randomly between states.

Gaussian Emissions

Each state emits observations from a multivariate Gaussian:

\[b_i(\mathbf{o}_t) = \mathcal{N}(\mathbf{o}_t; \boldsymbol{\mu}_i, \boldsymbol{\Sigma}_i)\]

The learned means and covariances for each state provide a probabilistic fingerprint of bull and bear markets.

Wavelet Transform: Seeing the Frequency Domain

Why wavelets? Because market activity isn’t just about magnitude—it’s about when and at what frequency patterns occur. I implemented the Continuous Wavelet Transform using the Complex Morlet wavelet:

\[\psi(t) = \frac{1}{\sqrt{\pi f_b}} \exp(2\pi i f_c t) \exp\left(-\frac{t^2}{f_b}\right)\]

The wavelet energy at each time step quantifies signal activity:

\[E_t = |W_t|^2\]

I engineered this as a rolling window feature—extracting the right-edge coefficient to avoid any lookahead bias. The result? A clean, forward-looking measure of market turbulence.

The Trading Strategy

Knowledge without action is worthless. I translated my regime detection into a systematic trading strategy with clear rules:

Regime-Aware Decision Matrix

Current Regime Signal Position Action
BULL BUY Flat ✅ ENTER LONG
BULL SELL Long ✅ EXIT
BEAR BUY Flat ❌ BLOCKED
BEAR SELL Long ✅ EXIT (graceful)

The key insight: block new entries in bear regimes, but allow graceful exits to avoid forced liquidation whipsaws.

Results: Testing Against the Financial Crisis

I trained my model on SPY data from 2000-2007, then tested it on the crucible of 2008—the global financial crisis. The ultimate stress test.

HMM Strategy (blue) vs Buy-and-Hold (orange) during the 2008 financial crisis. The shaded regions show detected market regimes.

What the Chart Shows:

  • Blue Line: My HMM strategy’s portfolio value
  • Orange Line: Passive buy-and-hold benchmark
  • Shaded Regions: Detected bull (green) vs bear (red) regimes
  • Markers: Entry and exit points based on regime changes

The strategy correctly identified the regime shift and reduced exposure during the worst of the crisis.

State Analysis: What My Model Learned

Statistical characteristics of the learned hidden states—showing return distributions, state frequencies, and transition probabilities.

Learned State Characteristics:

State Mean Return Volatility Interpretation
State 0 (BULL) Positive \(\mu_r^+\) Lower \(\sigma\) Calm uptrend
State 1 (BEAR) Negative \(\mu_r^-\) Higher \(\sigma\) Volatile decline

The model autonomously discovered the classic bull/bear dichotomy without any labeled training data.

Technical Implementation

I built this entire system in Python, implementing:

  • Forward Algorithm — \(O(N^2T)\) dynamic programming for likelihood computation
  • Backward Algorithm — Complementary pass for state occupancy probabilities
  • Viterbi Decoding — Maximum likelihood state sequence extraction
  • Baum-Welch (EM) — Iterative parameter estimation until convergence
  • Wavelet Feature Extraction — Rolling CWT with Complex Morlet kernel
  • Walk-Forward Backtesting — Strict train/test separation to avoid lookahead bias

Project Structure:

HMM_Multi/
├── 1_download_data.py    # Fetch SPY historical data
├── 3_train_hmm.py        # Train HMM with Baum-Welch
├── 4_backtest.py         # Regime-aware backtesting
├── 5_visualize.py        # Generate analysis plots
├── config.py             # Centralized parameters
├── wavelet_features.py   # CWT feature extraction
└── run_pipeline.py       # End-to-end execution

Explore the Full Implementation

This project demonstrates my understanding of:

  • Probabilistic graphical models and latent variable inference
  • Time-frequency analysis via wavelet transforms
  • Systematic trading with risk-managed position sizing
  • Scientific Python stack (NumPy, pandas, hmmlearn, PyWavelets)
  • Quantitative finance concepts (regime detection, backtesting, alpha generation)

Dive into the complete codebase with detailed mathematical documentation:

View Full Project on GitHub

Quick Start:

git clone https://github.com/Kunle-xy/OguntoyeProjects.git
cd "Hidden Markov Model/HMM_Multi"
pip install -r requirements.txt
python run_pipeline.py

The README includes comprehensive coverage of:

  • Complete mathematical derivations for all HMM algorithms
  • Wavelet transform theory and implementation details
  • State-observation diagrams and architecture visualizations
  • Configuration reference for customizing parameters
  • Strategy execution pipeline with risk management

Technical Skills Demonstrated

  • Hidden Markov Models and Expectation-Maximization
  • Wavelet transforms for time-frequency feature extraction
  • Gaussian mixture modeling and multivariate statistics
  • Quantitative backtesting with walk-forward methodology
  • Python scientific computing (NumPy, pandas, hmmlearn, PyWavelets)
  • Financial time series analysis and regime detection
  • Risk management and systematic trading strategies