Regime-Switching Jump Diffusion

Combines three distinct components into a single asset price process:

• Continuous Diffusion: A standard Brownian motion component capturing smooth, incremental price changes
• Jump Process: A compound Poisson process modeling sudden, discontinuous price moves (e.g., earnings surprises, flash crashes)
• Regime Switching: A latent Markov chain governing transitions between states, where each state has its own diffusion parameters, jump intensity, and jump size distribution

This tripartite structure allows the model to simultaneously capture volatility clustering, fat tails, and structural breaks that single-component models miss.

The arrival rate of jumps is not constant but varies with the prevailing market regime:

• High-volatility regimes typically exhibit elevated jump intensity (e.g., 5-10 jumps per year) with larger average magnitudes
• Low-volatility regimes show infrequent, smaller jumps (e.g., 0.5-2 jumps per year)
• The jump size distribution itself can be regime-conditional, often modeled as log-normal with state-dependent mean and variance

This feature addresses the empirical observation that market crashes cluster in turbulent periods rather than occurring uniformly across time.

Regime evolution follows a first-order Markov chain with a finite number of states (typically 2-4):

• Transition Probability Matrix: An N×N stochastic matrix where entry p_ij represents the probability of moving from regime i to regime j in the next time step
• Persistence: Diagonal elements are usually high (0.90-0.98), meaning regimes tend to persist once entered
• Ergodic Distribution: The long-run stationary probabilities derived from the transition matrix indicate the unconditional proportion of time spent in each regime

The Markov property ensures that future regime probabilities depend only on the current state, not the full history.

Calibration typically employs the Expectation-Maximization (EM) algorithm due to the latent nature of both the regime sequence and the jump occurrences:

• E-Step: Compute the smoothed probabilities of being in each regime at each time point, along with the probability that each observed return contains a jump component
• M-Step: Update all model parameters (diffusion volatilities, jump intensities, jump size moments, transition probabilities) to maximize the expected complete-data log-likelihood
• The Hamilton filter provides the recursive forward-backward inference of regime probabilities required within each EM iteration

Alternative Bayesian approaches using Markov Chain Monte Carlo (MCMC) allow full posterior inference with prior regularization.

The model provides a more realistic framework for derivatives pricing compared to Black-Scholes:

• Volatility Smile/Skew: The combination of jumps and regime switching naturally generates the implied volatility patterns observed in options markets without ad-hoc parameter adjustments
• Regime-Conditional Pricing: Option values can be computed conditional on the current inferred regime, producing state-dependent Greeks for dynamic hedging
• Risk-Neutralization: Requires specifying the market price of regime-switching risk and jump risk, typically through Esscher transform or equilibrium arguments
• Closed-form solutions are generally unavailable, necessitating Fourier transform methods or Monte Carlo simulation for pricing

Regime-switching jump diffusion enhances traditional risk metrics by conditioning on the inferred state:

• Regime-Conditional Value-at-Risk (Regime-CVaR): Tail risk estimates that automatically adjust upward during high-volatility, high-jump-intensity regimes
• Dynamic Hedging: Hedge ratios can be recomputed as regime probabilities shift, reducing exposure before anticipated turbulence
• Stress Testing: The transition matrix enables simulation of regime paths, including prolonged crisis scenarios with elevated jump activity
• Early Warning: A rising probability of transitioning to a high-jump regime serves as a quantitative signal for increasing market fragility

The foundational framework that extends geometric Brownian motion by adding a compound Poisson process to capture sudden, discontinuous price jumps. Key characteristics:

• Asset returns follow a continuous diffusion plus discrete jumps
• Jump sizes are typically modeled as log-normally distributed
• Jump arrivals follow a Poisson process with constant intensity λ
• Provides analytical solutions for European option pricing
• Captures the volatility smile observed in options markets
• Forms the base upon which regime-switching extensions are built

A stochastic counting process that models the random arrival of discrete events—in this context, price jumps—over continuous time. Essential properties:

• Jump intensity (λ) governs the expected frequency of jumps per unit time
• Inter-arrival times between jumps are exponentially distributed
• The number of jumps in non-overlapping intervals are independent
• In regime-switching extensions, λ becomes state-dependent, varying across bull, bear, and crisis regimes
• Enables modeling of clustered jump activity during high-volatility periods

A stochastic volatility model where volatility components operate at multiple frequencies, each governed by a Markov-switching process. Distinguishing features:

• Captures long-memory volatility persistence without long-range dependence in parameters
• Volatility is the product of multiple first-order Markov components with heterogeneous transition probabilities
• Naturally generates volatility clustering and fat-tailed return distributions
• Can be extended to incorporate jump components for extreme events
• Provides superior out-of-sample volatility forecasts compared to GARCH-class models
• Particularly effective for modeling currency and commodity markets

A class of continuous-time models where the drift, diffusion, and jump intensity are all affine (linear plus constant) functions of a state vector. Critical properties:

• The characteristic function has an exponentially affine form, enabling quasi-analytical pricing
• State variables can govern both diffusion volatility and jump arrival intensity simultaneously
• Regime-switching can be embedded by allowing the affine coefficients to depend on a hidden Markov state
• Widely used in credit risk modeling and sovereign debt pricing
• Duffie, Pan, and Singleton (2000) provide the canonical framework
• Enables efficient transform-based pricing of complex derivatives

Point processes where the occurrence of a jump increases the probability of subsequent jumps in the near future, creating temporal clustering. Key aspects:

• Jump intensity follows a stochastic differential equation that spikes after each event and decays exponentially
• Captures the empirical phenomenon of jump clustering during financial crises
• Can be combined with regime-switching to distinguish between normal clustering and crisis-level cascades
• Parameters include baseline intensity, excitation magnitude, and decay rate
• Originally developed for earthquake modeling, now applied to high-frequency trading and flash crash analysis
• Provides a more realistic alternative to the memoryless Poisson process

An extension where the underlying stochastic process follows a Lévy process whose parameters—including jump measure characteristics—switch according to a Markov chain. Core components:

• Lévy processes encompass both continuous Brownian motion and infinite-activity jump processes
• The Lévy measure determines the frequency and size distribution of jumps
• Regime-switching allows the entire Lévy triplet (drift, diffusion, jump measure) to change across states
• Captures both frequent small jumps (normal market noise) and rare large jumps (crisis events)
• Models like Variance Gamma and Normal Inverse Gaussian can be made regime-dependent
• Provides extreme flexibility for fitting options surfaces across different market conditions