Transition Probability Matrix

Each row of the matrix must sum to exactly 1, representing a complete probability distribution over all possible next states. For a 2-regime model with states {Bull, Bear}, if the probability of staying in Bull is 0.95, the probability of switching to Bear must be 0.05. This constraint ensures the model never creates or destroys probability mass:

• Row i represents the conditional distribution P(· | current state = i)
• The sum of probabilities for all possible transitions from state i equals 1
• This property is enforced during maximum likelihood estimation using constrained optimization

The diagonal entries p_ii represent the probability of remaining in the current regime. In financial applications, these values are typically high (often > 0.90), reflecting the empirical observation that market regimes are persistent:

• A Bull market tends to stay Bull; a Bear market tends to stay Bear
• The expected duration of regime i is calculated as 1 / (1 - p_ii)
• Example: If p_Bull,Bull = 0.95, the expected Bull regime lasts 20 periods
• Low diagonal values indicate a jumpy, unstable model that may overfit noise

For an irreducible and aperiodic transition matrix, a unique long-run equilibrium distribution π exists, satisfying π = πP. This vector represents the unconditional probability of being in each regime, independent of the starting state:

• π is the left eigenvector of P corresponding to eigenvalue 1
• It answers: 'What fraction of time does the market spend in each regime?'
• Computed by solving the linear system π(P - I) = 0 with the constraint Σπ_i = 1
• If π_Bull = 0.75, the market is in a Bull regime 75% of the time in the long run

An absorbing state has p_ii = 1.0, meaning once entered, the process can never leave. In financial modeling, this is typically undesirable as it implies a permanent regime with no switching:

• A transition matrix with absorbing states is reducible
• Reducible matrices may have multiple stationary distributions depending on initial conditions
• In practice, constraints are often imposed to ensure all p_ii < 1
• Near-absorbing states (e.g., p_ii = 0.999) can cause numerical instability during estimation

In TVTP models, the transition matrix is no longer constant but becomes a function of exogenous covariates z_t. The probabilities are typically parameterized using a logistic or probit link function:

• p_ij(z_t) = exp(α_ij + β_ij·z_t) / Σ_k exp(α_ik + β_ik·z_t)
• This allows regime switches to depend on observable variables like the VIX, credit spreads, or macroeconomic indicators
• The matrix remains row-stochastic by construction through the softmax normalization
• TVTP models capture the intuition that transitions are more likely during periods of market stress

The second-largest eigenvalue λ_2 of the transition matrix governs the mixing rate—how quickly the process converges to its stationary distribution:

• λ_1 = 1 always (corresponding to the stationary distribution)
• |λ_2| close to 1 indicates slow mixing and high persistence
• The mixing time is approximately -1 / log(|λ_2|)
• A spectral gap (1 - |λ_2|) near zero signals near-reducibility and potential identification problems
• This decomposition is used diagnostically to assess whether a fitted model can reliably distinguish between regimes

The long-run unconditional probability of being in a specific regime, derived directly from the transition probability matrix. It represents the expected proportion of time the market spends in each state.

• Calculated by solving π = πP, where P is the transition matrix
• For a two-state bull/bear model, ergodic probabilities might show the market is in a bull regime 65% of the time
• Assumes the Markov chain is irreducible and aperiodic
• Provides the steady-state distribution independent of the starting regime

An extension where the probability of moving between regimes depends on observable exogenous variables rather than remaining constant. This allows the transition matrix to respond dynamically to economic conditions.

• Transition probabilities become functions of macroeconomic indicators, volatility indices, or credit spreads
• A logistic function often links covariates to probabilities: p(t) = 1 / (1 + exp(-β'x_t))
• Captures the intuition that regime switches are more likely during high VIX environments
• Requires careful variable selection to avoid overfitting the switching dynamics

A special case of the Expectation-Maximization (EM) algorithm used to estimate the parameters of a Hidden Markov Model, including the transition probability matrix, when the state sequence is unknown.

• Iteratively refines estimates of the transition matrix, emission parameters, and initial state distribution
• The E-step computes expected state occupancy and transition counts using forward-backward probabilities
• The M-step re-estimates the transition matrix by normalizing expected transition counts
• Converges to a local maximum of the likelihood function; multiple random restarts are standard practice

A measure of systematic risk that varies depending on the prevailing market regime, acknowledging that a stock's sensitivity to the market index differs in bull versus bear phases.

• Estimated by conditioning the CAPM beta on the latent regime: β_s for each state s
• A defensive stock might have β_bear = 0.6 and β_bull = 1.1
• The transition matrix governs how frequently the portfolio's risk profile shifts
• Enables dynamic hedging strategies that adjust exposure based on the inferred regime probability

A dynamic programming algorithm that decodes the most likely sequence of hidden states (regimes) given a sequence of observed market data and a fitted model including the transition matrix.

• Maximizes the joint probability of the entire state sequence, not just marginal probabilities at each time step
• Uses a trellis structure to recursively compute the highest probability path
• Essential for regime classification: labeling each historical period as bull, bear, or sideways
• Provides the globally optimal state sequence under the estimated model parameters

A tail-risk measure that calculates expected loss conditional on exceeding the Value-at-Risk threshold, with the loss distribution specifically modeled for the current market regime using the transition matrix.

• CVaR in a high-volatility regime will be significantly larger than in a low-volatility regime
• The transition matrix provides the probability weights for blending regime-specific risk estimates
• Forward-looking CVaR can be computed by simulating future regime paths using the estimated transition probabilities
• Addresses the failure of static risk models during regime shifts and correlation breakdowns