[Paper Review] Risk Measures in a Regime Switching Model Capturing Stylized Facts

Rainer Haidinger, Richard Warnung|arXiv (Cornell University)|Dec 17, 2012

Stochastic processes and financial applications3 references3 citations

TL;DR

This paper proposes a regime-switching model using a two-state Markov chain and generalized hyperbolic distributions to capture stylized facts in financial returns, such as fat tails, volatility clustering, and skewness. It applies Fourier transform methods—adapted from option pricing—to efficiently compute Value-at-Risk (VaR) and Expected Shortfall (ES), demonstrating superior out-of-sample performance compared to a simple GHYP model with 10 VaR breaches versus 36 in 610 days.

ABSTRACT

We pick up the regime switching model for asset returns introduced by Rogers and Zhang. The calibration involves various markets including implied volatility in order to gain additional predictive power. We focus on the calculation of risk measures by Fourier methods that have successfully been applied to option pricing and analyze the accuracy of the results.

Motivation & Objective

• To develop a risk model that captures key stylized facts of financial returns, including fat tails, volatility clustering, and skewness.
• To enhance predictive power by incorporating forward-looking implied volatility (VDAX) into the calibration of a regime-switching model.
• To apply Fourier transform techniques from option pricing to efficiently compute risk measures like VaR and ES in a non-Gaussian, regime-switching framework.
• To evaluate the model’s performance through in-sample and out-of-sample testing, comparing it to a simpler GHYP-based benchmark.

Proposed method

• The model uses a hidden two-state Markov chain to represent economic regimes, with returns conditionally distributed as generalized hyperbolic distributions (GHYP) in each state.
• Calibration is performed via maximum-likelihood estimation using 3 years of data from equity, bond, commodity, and VDAX indices to improve predictive accuracy.
• Fourier transform methods are employed to derive closed-form expressions for VaR and ES, enabling fast computation via the Fast Fourier Transform (FFT).
• The characteristic function of the return distribution is derived under the regime-switching framework, allowing for numerical inversion to compute risk measures.
• Posterior state probabilities are updated dynamically using filtering techniques, enabling time-varying risk estimates.
• Expected shortfall is computed using an integral formula involving the characteristic function and a complex shift parameter to ensure convergence.

Experimental results

Research questions

• RQ1 Can a regime-switching model with generalized hyperbolic distributions effectively capture the stylized facts of financial returns, such as fat tails and volatility clustering?
• RQ2 Does including implied volatility (VDAX) in the calibration improve the predictive power of the risk model?
• RQ3 Can Fourier transform techniques from option pricing be successfully adapted to compute VaR and ES in a non-Gaussian, regime-switching framework?
• RQ4 How does the performance of the regime-switching model compare to a simpler GHYP-based model in out-of-sample risk forecasting?

Key findings

• The regime-switching model with VDAX calibration achieved 10 VaR breaches in 610 out-of-sample days, yielding a p-value of 0.99 in the binomial test, indicating reliable risk estimation.
• The runs test for breach mixing yielded a p-value of 0.6772, suggesting no systematic pattern in VaR breaches, supporting the model’s validity.
• Expected shortfall estimates dynamically adjusted between two state-specific levels, reflecting changing market regimes as captured by posterior probabilities.
• The model outperformed a simple GHYP-based benchmark, which recorded 36 VaR breaches in the same period, indicating superior responsiveness to market conditions.
• The use of FFT-enabled Fourier methods allowed for efficient and accurate computation of risk measures, validating their applicability beyond option pricing.
• Despite strong performance, the model’s sensitivity to calibration window length was identified as a key limitation, highlighting the need for regular recalibration.