Markets do not announce the regime they are in.

We observe prices, returns, and volatility, but the underlying state driving those observations, bull market, bear market, crisis, or recovery, is hidden. We have to infer it indirectly from noisy data.

Hidden Markov Models extend Markov chains by introducing these latent states. Instead of observing the state directly, we observe signals generated by it and use probability to infer which regime is most likely active.

This simple extension makes HMMs one of the most widely used tools for regime detection, speech recognition, and sequential inference. When the state cannot be observed directly, Hidden Markov Models provide a principled way to estimate it.

📚 Model Definition

This following structure is an example of a Markov chain...

At a high level, a Hidden Markov Model has two layers: hidden states (S₁, S₂, S₃, S₄) and observations (O₁, O₂, O₃, O₄).

The hidden states evolve according to a Markov chain. The transition matrix A determines the probability of moving between states, with entries such as A₁₂, A₂₃, and A₃₄ describing the likelihood of transitioning from one hidden regime to the next.

Each hidden state then generates an observation through the emission model E. The emission probabilities (E₁, E₂, E₃, E₄) describe how likely each hidden state is to produce the observations we actually measure.

The key challenge is that we never observe Sₜ directly. We only observe Oₜ and use the transition probabilities A together with the emission probabilities E to infer the most likely sequence of hidden states.

In other words, an HMM separates the latent process (Sₜ) from the observed data (Oₜ), allowing us to infer hidden regimes from noisy observations.

📈 Model Applications

In practice, Hidden Markov Models are widely used for market regime detection. 

Rather than assuming markets behave the same way through time, HMMs infer latent states such as bull markets, bear markets, low-volatility, and crisis regimes from observable returns, volatility, or macroeconomic data.

They are also used in statistical arbitrage, volatility modeling, speech recognition, and anomaly detection, wherever the underlying process cannot be observed directly but leaves measurable signals.

The challenge is that the hidden states are only as useful as they are stable.

Markets evolve. Relationships change. A regime learned from one decade may not correspond to the same economic environment in the next. As a result, the estimated transition probabilities, emission distributions, and even the interpretation of the hidden states can drift over time.

This is the fundamental limitation of HMMs. They infer regimes from historical data, but there is no guarantee those regimes remain stationary. The model can uncover hidden structure, but whether that structure persists is ultimately an empirical question.

🎓 A Little Story

💡 Takeaway

Hidden Markov Models are powerful because they acknowledge something most models ignore: the system driving the data may itself be hidden.

But inferring hidden structure is not the same as discovering permanent truth. If the underlying regimes are unstable, the inferred states will be unstable too. That is not necessarily a failure of the model—it may be an honest reflection of the data.

The real lesson is that models cannot create structure where none exists. HMMs are excellent at uncovering latent patterns, but they cannot guarantee those patterns will persist. In quantitative finance, that distinction is often more important than the model itself.

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The following Markov chain is constructed with an absorbing state and the probability is observed applying the Chapman-Kolmogorov result.