Regime-Switching Stochastic Volatility Model

Stochastic Vol

A Markov-modulated Heston model where all parameters evolve according to a hidden Markov chain representing market regimes (e.g., calm vs crisis). The model captures regime-dependent volatility dynamics, different drift and correlation per regime, and the persistence of each market state.

Mathematical Formulation

Z_{t} \in {0, 1, \dots, N - 1} follows Markov chain with transition matrix P

d S_{t} = μ_{Z_{t}} S_{t} d t + v_{t} S_{t} d W_{t}^{S}

d v_{t} = κ_{Z_{t}} (θ_{Z_{t}} - v_{t}) d t + ξ_{Z_{t}} v_{t} d W_{t}^{v}

Cov (d W_{t}^{S}, d W_{t}^{v}) = ρ_{Z_{t}} d t

Parameters

Symbol	Description	Constraint
$Z_{t}$	Current regime state	$Z_{t} \in {0, ..., N - 1}$
$P$	Transition probability matrix	$\sum_{j} P_{ij} = 1$
$μ_{i}$	Drift in regime i	$μ_{i} \in R$
$κ_{i}$	Mean reversion speed in regime i	$κ_{i} > 0$
$θ_{i}$	Long-run variance in regime i	$θ_{i} > 0$
$ξ_{i}$	Vol-of-vol in regime i	$ξ_{i} > 0$
$ρ_{i}$	Correlation in regime i	$ρ_{i} \in [- 1, 1]$

Key Assumptions

•Market operates in discrete regimes (calm, crisis, etc.)
•Parameters switch according to hidden Markov chain
•Captures regime-dependent dynamics
•Persistence in each regime state
•Extends Heston with Markov regime switching

Reference

Hamilton, J.D. (1989). "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle." Econometrica.

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Regime-Switching Stochastic Volatility Model

Mathematical Formulation

Parameters

Key Assumptions

Reference

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Heston

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