1. Introduction
1.1. Regime Switching
Financial markets do not evolve in a smooth, linear fashion. Instead, they transition through distinct regimes, which can be understood as extended periods during which market behaviour remains relatively homogeneous. These regimes may feature persistent positive or negative returns, different volatility conditions, or sustained risk-on and risk-off dynamics. A regime switch represents a sudden and pronounced break in these patterns, often triggering large reallocations of capital across asset classes and sharp market moves. In one regime, equity returns may be strong, volatility subdued and credit spreads tight; in another, drawdowns cluster, correlations spike and even traditionally “safe” assets become unstable. For investors, traders, and risk managers, detecting these regimes in real time is crucial: the same portfolio can behave very differently depending on whether the market is in a “risk-on” or “risk-off” environment.
Regime-type thinking has become increasingly important as markets have repeatedly experienced considerable regime switches in the past decade. The COVID-19 crash in early 2020 compressed an entire bear market into a matter of weeks, followed by an unprecedented policy-driven rebound. The 2022 inflation shock and rate-hiking cycle marked a violent break from the post-2008 era of low rates and ample liquidity, driving the worst joint performance of stocks and bonds in decades and repricing term premia across curves. More recently, the AI-driven equity rally from late 2022 onward has signalled a different type of regime again, one which suggests a seeming divergence between equity markets and underlying market conditions. Each of these episodes illustrates the same underlying point: the distribution of returns is not stationary. Structural breaks in macro conditions (pandemics, inflation shocks, policy shifts, technology narratives) show up as regime changes in prices. One of the most important patterns that characterises relevant regime shifts is the distinction between risk on and risk off behaviours. Ceteris paribus, in risk-on regimes, investors are willing to bear more risk: they rotate into higher-beta assets, compress credit spreads, sell volatility, and accept lower compensation for duration and liquidity risk. In risk-off regimes, the opposite occurs: demand for safety and liquidity rises, risk premia widen, and hedging activity increases.
1.2. Example: Swap Spreads
Focusing on classic risk-on vs. risk-off dynamics in regime switches specifically, they are conveniently illustrated through swap spreads. Swap spreads reflect the difference in the par rate of a vanilla interest rate swap with the same maturity as an underlying government bond, say a treasury bond. Whilst both legs are driven by general interest rate and inflation expectations, interest rate swaps reflect flows from pension funds, mortgage hedging and for the sake of this discussion, most importantly, counterparty risk. In a textbook risk-off episode, investors may rush into government bonds as safe assets, pushing their yields down relative to swap rates and causing swap spreads to widen. In more benign, risk-on conditions, demand for duration in government bonds can fade or balance with paying flows in swaps, leading to tighter or even negative swap spreads in certain maturities.
Major risk-on and risk-off catalysts have dominated market dynamics over the past five years: the COVID shock and subsequent QE stimulus followed by unprecedented Treasury issuance; the escalation of the Russia–Ukraine conflict; the Silicon Valley Bank panic; and, more recently, the Trump tariff cycle. However, in real-life, the impacts are often less trivial:
The SVB episode behaved like a textbook risk-off shock: a rapid flight into Treasuries drove yields sharply lower, widening swap spreads (i.e., making them less negative). In contrast, the Russia-Ukraine conflict and the Liberation Day exhibited more mixed effects. Both episodes were risk-off in nature, but in each case the usual safe-haven yield rally was offset by competing forces, most notably the US inflation cycle from 2022-2023, and reflation concerns from Liberation Day, both of which exerted upward pressure on yields and compressed swap spreads.
1.3. Example: Equity Volatility
Equity markets themselves exhibit strong and recognisable regime-dependent behaviour. In risk-on regimes, equity indices typically show persistent upward trends, volatility remains compressed, and market breadth improves as a larger share of stocks participate in the rally. Factor leadership becomes stable: cyclical and high-beta names outperform, dispersion falls, and earnings revisions turn consistently positive. In risk-off regimes, the pattern flips: returns become negatively skewed, volatility spikes, and correlations across stocks converge sharply toward one, causing diversification to break down. Drawdowns cluster, sector rotations accelerate, and defensive segments, such as utilities, staples, and low-vol, take leadership while cyclical sectors underperform. Even microstructure indicators shift, with wider intraday ranges, heavier tail events, and more erratic market depth. These dynamics highlight that equity markets alone contain rich information about changing regimes, with volatility, correlations, and breadth acting as early warning signals of transitions from stability to stress.
Equity volatility offers another clear window into regime behaviour. The VIX and its European counterpart, the V2X, tend to remain compressed for long periods during risk-on environments, reflecting stability, strong liquidity, and confidence in forward earnings. Risk-off regimes, in contrast, are marked by sharp volatility spikes as markets rapidly reprice uncertainty. Episodes such as the COVID crash, the 2022 inflation shock, the SVB panic, and the recent tariff-driven volatility cycle each produced abrupt surges in implied volatility, followed by extended normalisation. These shifts make volatility indices particularly effective for highlighting regime breaks: they respond quickly, capture changes in sentiment, and often turn before broader equity indices. The chart below illustrates how VIX and V2X have evolved across recent market regimes, with volatility clustering around major macro shocks and stabilising in periods of renewed risk appetite.
This raises a natural question: rather than labelling such episodes as “risk-off regimes” only in hindsight, can we build a model that statistically infers adverse market conditions directly from observable variables? In other words, can regime-identification models systematically detect shifts in market behaviour at the time they occur, without relying on ex-post classification?
2. Literature Review
The concept of regime-switching was introduced by Hamilton (1989), with the Markov regime-switching model to explain cyclical changes in macroeconomic and financial time series. His framework formalised the idea that asset returns follow distinct processes across hidden market states, such as expansions and recessions. Later studies, including Bulla et al. (2011) and Nystrup et al. (2015, 2020b), extended these models into investment strategy design, using regime signals from Hidden Markov Models to time exposure between risky and risk-free assets and emphasising the importance of regime persistence, not changing erratically in response to short-term fluctuations. A major methodological advance came from Bemporad et al. (2018), who introduced the Statistical Jump Model (JM), which penalises excessive regime switching to yield more stable, persistent signals. Shu, Yu, and Mulvey (2024) further adapted this framework to dynamic portfolio management, demonstrating that JM-guided strategies outperform traditional approaches by reducing downside risk and improving risk-adjusted returns.
2.2. Practical Application of Regime-Switching Models
Regime-switching models are incorporated into investment strategy design through a structured process that begins with identifying the prevailing market regime using models such as the Hidden Markov Model (HMM) or the Statistical Jump Model (JM). Once a regime is identified, the key consideration becomes whether that regime is likely to persist, requiring a balance between responsiveness to new information and resistance to short-term noise. After reliable regime detection, the information is translated into portfolio actions that adjust exposure according to the inferred market environment. In practice, regime-switching models are most commonly applied in asset allocation, market timing, and risk management.
Asset allocation
In asset allocation, portfolio weights are adjusted based on the inferred market regime. The underlying intuition is straightforward: investors increase exposure to risky assets during favourable regimes (bull or low-volatility periods) and shift toward risk-free assets in unfavourable regimes (bear or high-volatility periods). Shu et al. (2024) apply this concept using the “0/1 strategy”, where the entire portfolio alternates between a risky and a risk-free asset depending on the identified regime. The simplicity of this binary rule isolates the informational value of the regime signal itself, providing a clear and interpretable test of how accurately the model distinguishes between stable and turbulent market conditions.
Market timing
Regime-switching models also support market timing by offering a systematic mechanism to reduce exposure ahead of downturns and re-enter risky assets during recoveries. Rather than forecasting future prices, these models infer the current unobserved market state from observable characteristics such as volatility or downside deviation. The objective is not to predict regime shifts but to recognise when a shift has occurred and then benefit from the persistence of the new state. The Statistical Jump Model enhances this process by imposing a jump penalty that discourages excessive switching, ensuring that regime signals remain stable and economically meaningful. Persistence of this kind reduces trading frequency and turnover, improving implementability while preserving responsiveness to market transitions. During periods of stress, such as the 2008 financial crisis or the 2020 COVID-19 market crash, regime-switching strategies that employed JM-based signals successfully exited risky assets, limiting drawdowns and re-engaging during subsequent recoveries.
Risk management
In risk management, regime-switching models capture the asymmetric nature of market risk, where adverse regimes are typically associated with higher volatility, lower expected returns, and stronger cross-asset correlations. By identifying and responding to these transitions, regime-aware strategies can mitigate downside risk while preserving long-term growth potential. Empirical evidence shows that strategies guided by persistent regime identification achieve lower volatility, reduced drawdowns, and improved risk-adjusted returns compared with static or more reactive approaches. Moreover, the persistence embedded in models like the JM provides robustness against practical frictions such as transaction costs and trading delays, allowing signals to remain effective even when implementation is imperfect.
2.3. Evolution from Hidden Markov Models to Statistical Jump Models
Traditional Hidden Markov Models (HMMs) have long served as the standard framework for regime identification in finance. These models assume that observed returns are generated by an unobserved Markov chain of latent states, with each regime defined by distinct statistical properties, typically differing in volatility or mean return. However, HMMs tend to be sensitive to noise, resulting in overly frequent switches that undermine their practical use in trading and asset management. To address these challenges, various non-parametric, data-driven alternatives like trend filtering and special clustering HMMs have emerged. This article focuses on the application of jump models as one such alternative. The Statistical Jump Model (JM) addresses these limitations by introducing a jump penalty; a regularisation term that penalises excessive transitions between regimes. This enhancement encourages state persistence and yields more stable regime sequences, reducing the risk of false signals. The model also allows for richer feature sets, incorporating both risk and return measures (e.g., downside deviation and Sortino ratios), rather than relying solely on volatility as in HMMs.
2.4. Regime-Switching Strategy Pipeline
The regime-switching strategy follows a four-stage pipeline comprising (1) regime identification, (2) regime persistence and forecast assessment, (3) trading and allocation design, and (4) out-of-sample evaluation. The first stage applies unsupervised learning to infer latent market states from return- and risk-based features using the Statistical Jump Model (JM). The second stage relies on the persistence of inferred regimes rather than explicit forecasting, assuming that the most recently identified state is likely to continue in the near term. In the third stage, regime information is translated into portfolio decisions through an interpretable “0/1 strategy,” which allocates fully to the risky asset during favourable regimes and to the risk-free asset during unfavourable ones. Finally, the framework is evaluated out-of-sample across major equity indices (including S&P 500, DAX, and Nikkei 225) from 1990 to 2023, accounting for transaction costs and trading delays to reflect realistic investment conditions.
3. Methodology
3.1. Hidden Markov Models
To formally introduce Hidden Markov Models (HMMs), we define two stochastic processes: a hidden state sequence $\{S_t\}_{t=1}^T$, where $S_t \in \{1, \dots, K\}$, and an observed sequence $\{Y_t\}_{t=1}^T$, where $Y_t \in \mathbb{R}^d$. The hidden process governs the dynamics of the system, while the observations are conditionally dependent on the hidden states.
The model is parameterized by:
$$\Theta = (\pi, A, \Phi)$$
Where:
- $\pi_i = p(S_1 = i)$ denotes the initial state probabilities, i.e. the probability of starting in each hidden state at time $t = 1$.
- $A = (a_{ij})$ is the transition matrix, with $a_{ij} = P(S_t = j \mid S_{t-1} = i)$.
- $\Phi = \{\phi_k\}$ for $k = 1, \dots, K$ are the emission parameters, defining the conditional distributions $$p(Y_t \mid S_t = k; \phi_k) = f(Y_t; \phi_k)$$
These parameters are represented or estimated through the emission matrix $B = (b_{ij})$, where $b_{ij} = p(y_t = j \mid s_t = i)$.
Under the first-order Markov assumption:
$$p(S_t \mid S_{t-1}, S_{t-2}, \dots, S_1) = p(S_t \mid S_{t-1})$$
which states that the probability of being in a certain hidden state at time □ depends only on the hidden state at time □ − 1, and conditional independence of observations, the joint distribution of the hidden and observed sequences factorizes as:
$$ p(Y_{1:T}, S_{1:T} \mid \Theta) = \pi_{S_1} \prod_{t=2}^T a_{S_{t-1}, S_t} \prod_{t=1}^T b_{S_t}(y_t)$$
Because the joint likelihood of a Hidden Markov Model involves products of many small probabilities, it is standard to work with log-probabilities for numerical stability and analytic simplicity. That is,
$$\log p(Y, S \mid \Theta) = \log \pi_{S_1} + \sum_{t=2}^T \log a_{S_{t-1},S_t} + \sum_{t=1}^T \log b_{S_t}(y_t)$$
Since the logarithm is a monotonic function, maximizing the log-likelihood is equivalent to maximizing the likelihood itself, but the log form converts products into sums.
The parameters Θ are typically estimated by maximizing the marginal likelihood. That is usually done by the Baum-Welch algorithm, a special case of the Expectation-Maximisation (EM) for HMMs. It alternates between computing expectations over hidden states (E-step) using the forward-backward algorithm and maximising those expected values (M-step).
However, because the likelihood function of an HMM is non convex, the Expectation-Maximization algorithm is not guaranteed to find the global maximum, and the resulting estimates depend on initialization. This sensitivity to starting values often requires multiple random restarts or prior constraints to obtain stable parameter estimates.
Once the model parameters Θ are estimated, the most probable sequence of hidden states is obtained using the Viterbi algorithm, which solves
$$\hat{S}_{1:T} = \arg\max_{S_{1:T}} p(S_{1:T} \mid Y_{1:T}, \hat{\Theta})$$
In contrast to Baum-Welch, which computes expected state probabilities, the Viterbi algorithm yields a single maximum a posteriori (MAP) path.
HMMs are parametric models, meaning they assume that
$$
p(y_t \mid S_t = k) = f(y_t;\,\phi_k)
$$
i.e. they belong to a known family of distributions f (e.g. Gaussian) governed by a parameter □□ such as mean or variance. For instance, a parametric HMM might assume
$$
Y_t \mid S_t = k \sim \mathcal{N}(\mu_k, \sigma_k^2)
$$
and if this assumption is wrong, this will lead to a conditional distribution misspecification, falling within a frame in which the model compensates for its poor fit by artificially switching states to fit what is included in regime variation.
A similar problem fits the framework of sojourn time (or dwell time), i.e. the time spent in a specific regime before switching. Formally,
$$P(D_i = d) = (a_{ii})^{d-1} (1 - a_{ii})$$
This is a geometric distribution on the duration □, which implicitly defines that the probability of exiting regime decays exponentially over time. Immediate comment is that markets’ durations do not work empirically on geometric distributions, showing yet another limit of HMMs. When you force an HMM to use a geometric sojourn time, it can’t reproduce long regimes naturally, tending to switch too often.
These limitations motivate the use of more general frameworks (such as JMs) to formulate the problem as a direct optimization of a loss function involving both model parameters and model sequences, relaxing the strict probabilistic assumptions of HMMs.
3.2. Statistical Jump Models
Motivation and core idea
The Statistical Jump Model (JM) generalizes HMMs by replacing stochastic state transitions with a penalized segmentation framework. Instead of modelling transitions probabilistically, JMs directly penalize changes between consecutive regimes through a cost function. This transforms the regime detection problem from one of probabilistic inference to one of regularized optimization.
At its core, the JM introduces a jump penalty λ that discourages frequent switches between regimes. Let $\theta_t$ represent the regime parameters (for instance, the local mean and volatility of returns at time t). Given a sequence of observed returns $y_{1:T}$, the model estimates the most likely parameter sequence $θ_{1:T}$ by minimizing the global cost:
$$J(\theta_{1:T}) = - \sum_{t=1}^T \log p(y_t \mid \theta_t) \; + \; \lambda \sum_{t=2}^T \mathbf{1}_{\theta_t \neq \theta_{t-1}}$$
The first term ensures statistical fit to the data (likelihood), while the second term penalizes jumps between regimes.
The jump penalty $\lambda > 0$ controls the persistence of regimes:
- Small $\lambda$: the model permits frequent jumps, adapting rapidly but risking overfitting;
- Large $\lambda$: the model enforces smoother regime sequences, identifying only structural changes.
This formulation can be interpreted as a maximum a posteriori (MAP) problem under a prior that favours piecewise-constant regime paths.
Inference via dynamic programming
The optimization of $J(θ_{1:T})$ can be solved efficiently using dynamic programming. Define $C_t(k)$ as the minimal cost of explaining data up to time $t$ if the system is in regime $k$ at time $t$:
$$C_t(k) = -\log p(y_t \mid \theta_k) + \min \big( C_{t-1}(k),\; \min_{j \neq k} [C_{t-1}(j) + \lambda] \big)$$
This recursion implies that at each step, the model decides whether it is cheaper (in terms of cost) to stay in the same regime or to “jump” to a different one, paying the penalty $\lambda$. The dynamic programming solution proceeds forward in time, storing the minimal cost for each regime, and can be traced backward to recover the optimal sequence $\theta⋆_{1:T}$ .
Statistical and economic interpretation
From a probabilistic perspective, the Jump Model replaces the discrete Markov transition probabilities $A_{ij}$ with an implicit prior on regime persistence. This can be interpreted as an $\ell_0$-regularization on the number of jumps:
$$\sum_{t=2}^T \mathbf{1}_{\theta_t \neq \theta_{t-1}}$$
analogous to how sparsity penalties control model complexity in regression. Hence, the JM imposes sparsity in time rather than in parameters: it favours a small number of persistent regimes that explain the data well.
In financial terms, the estimated path $\theta⋆_{1:T}$ identifies stable periods of homogeneous behaviour, for instance, a low-volatility “risk-on” phase and a high-volatility “risk-off” phase. The parameter λ acts as a time-scale selector: low values detect short-term fluctuations, while high values reveal only major structural breaks, such as crises or policy shifts.
4. Model Implementation
In this section, we implement the Statistical Jump Model (JM) for regime detection using daily S&P 500 data from 2000–2025. First, we use observable return- and risk-based features to infer persistent bull and bear regimes in equity markets. Second, we translate these inferred regimes into systematic trading strategies and evaluate their performance out-of-sample.
4.1. Data and Features
Following Bemporad et al. (2018) and Shu et al. (2024), we construct a feature matrix that captures the multi-horizon behaviour of returns and downside risk. Bull and bear markets differ not only in average returns, but also in persistence, asymmetry, and the magnitude of negative deviations.
We compute rolling features over windows of 5, 20, and 60 trading days. These include:
Rolling Simple Returns
$$
r_t^{(w)} = \frac{P_t}{P_{t-w}} - 1
$$
Downside Deviation (for Sortino Ratio)
$$
\sigma_t^{-}(w) = \sqrt{\frac{1}{w}\sum_{i=1}^w \min(r_{t-i}, 0)^2}
$$
Sortino Ratio
$$
\text{Sortino}_t^{(w)} = \frac{\bar{r}_t^{(w)}}{\sigma_t^{-}(w)}
$$
Drawdown (classical)
$$
DD_t^{(w)} = \frac{P_t}{\max(P_{t-w:t})} - 1
$$
Log-Drawdown (used in JM)
$$
DD\_{\log,t}^{(w)} = \ln(1 + DD_t^{(w)})
$$
Log-drawdowns emphasize persistent declines more than short sharp drops, making them particularly suitable for regime models. In total, these produce a 9-dimensional feature vector per day: 3 returns, 3 Sortino ratios, 3 log-drawdowns. We intentionally exclude volatility indices, price levels, and macro variables to isolate the explanatory power of the JM itself.
4.2. Model Fitting and Jump Penalty Selection
We train the JM on data from 2000–2015 using the constructed features. The JM is governed by a single hyperparameter: the jump penalty $\lambda$, which controls regime persistence.
We evaluate:
$$
\lambda \in \{0, 5, 10, 20, 30, 40, 50, 60, 80, 100, 150\}
$$
For each value of $\lambda$ we:
- Fit the model in-sample
- Perform online out-of-sample inference
- Construct a 0/1 risk-on–risk-off strategy
- Compute the Sharpe ratio
A local peak occurs at $\lambda = 5$, where the model catches nearly every minor downturn, which temporarily boosts Sharpe by avoiding several small losses. However, this reactivity is unstable: it generates many false positives, overfits noise, and does not generalise well out-of-sample.
A global peak occurs around $\lambda = 60$–$80$, producing a persistent regime structure consistent with the empirical duration of real bear markets, macroeconomic cycle lengths, and the original JM literature.
4.3. Feature Importance
To understand why the JM classifies certain periods as bull or bear, we compute the regime centroids, the mean feature values in each regime:
$$
\mu_{\text{bull}} = \mathbb{E}[X_t \mid \text{bull}], \qquad
\mu_{\text{bear}} = \mathbb{E}[X_t \mid \text{bear}]
$$
These centroids represent the “average bull” and “average bear” market in feature space.
Because the JM itself has no internal feature weights (unlike a logistic regression), the best proxy for feature importance is the magnitude of separation (absolute difference) between centroids:
$$
\text{Importance}(i) = \left| \mu_{\text{bull}}^{(i)} - \mu_{\text{bear}}^{(i)} \right|
$$
This measures how strongly a feature differs between market states. A large difference means the feature plays a major role in separating regimes.
Long-horizon Sortino (60d) is the strongest separator, indicating persistent downside risk is a dominant signature of bear markets. Medium-horizon returns (20–60d) also differ sharply: bear regimes tend to cluster around extended negative drift. Log-drawdowns increase consistently in bears, reflecting structural market stress.
This confirms that the JM is not simply reacting to short-term volatility, but capturing deeper risk-return patterns.
4.4. In-Sample Fit
Using $\lambda = 80$, we plot the inferred in-sample S&P 500 regimes from 2000-2015:
The inferred regimes align closely with major historical episodes: Dot-com (2000–2002), GFC (2008–2009). “Micro-regimes” (short-lived bear pockets) also appear during volatility spikes (2011, 2015), but remain reasonably stable due to the jump penalty.
A key strength is regime persistence: unlike HMMs, which often flicker between states, the JM produces long, coherent episodes due to the jump penalty.
4.5. Out-of-Sample Online Inference
Next, we test the model on data from 2016-2025. We simulate online inference, where the model must classify regime $t + 1$ using only information up to time $t$. This replicates real-time usage of the JM in asset allocation.
At each day $t$:
- Observe features $X_{1:t}$
- Infer regime probabilities $z_t$
- Predict next-day regime:
$$ \hat{z}_{t+1} = \arg\max_z P(z_{t+1} \mid X_{1:t}) $$
The model successfully identifies the 2020 COVID crash and the late-2021 to 2022 inflation shock as a prolonged bear regime, as well as the March 2023 SVB panic. Notably, the JM also infers a bear regime starting in April 2025, following the introduction of new tariff policies by U.S. President Donald Trump. Despite being unsupervised, the inferred regimes align closely with economic reality.
5. Strategies and Results
With regimes estimated, we construct several trading strategies:
- S&P 500 Buy & Hold
- 0/1 Strategy: buy the S&P 500 during bull regimes, stay out during bear regimes.
- Regime-tilt strategies: replace risk-off leg with 3 Month T-Bills, Gold, or long-duration Treasuries (TLT, iShares 20+ Year Treasury Bond ETF).
Our implementation does not account for trading delays and transaction costs.
The out-of-sample equity curves show that regime-guided strategies behave defensively during JM-identified bear periods.
The Gold-tilt strategy achieves the highest cumulative return (16.8% annualised) and Sharpe ratio (≈1.16),, benefiting from gold’s strong hedging properties during bear periods. The other strategies, with the exception of Long Duration Bonds, all achieve a higher Sharpe than buy-and-hold. The 0/1 model boosts Sharpe from 0.77 to 0.93.
Importantly, all strategies reduce volatility, with the 0/1 and T-bill strategies reducing it from 17.2% to 11.5%. Maximum drawdown improves sharply, from –33.9% to –17.6% in the 0/1 strategy.
All regime-based strategies improve Expected Shortfall (5%) relative to buy-and-hold, highlighting the JM’s effectiveness at sidestepping fat-tailed downside.
6. Conclusions
The Statistical Jump Model proves to be a powerful and interpretable framework for market regime detection and tactical asset allocation. It successfully captures economically meaningful regimes, improves risk-adjusted returns, generalises well and reacts sensibly to new stress episodes.
Overall, the JM offers a data-driven, persistent, and interpretable alternative to complex black-box machine learning models. Its transparency and robustness make it particularly appealing for risk management, tactical positioning, and macro-aware investment processes.
Written by Caterina Molinari, Thomas Agosti, Vincenzo Volpe, Marc Weigand, Ilse Russell-Jones, Neil Pinto