Modeling Financial Stability Under Economic and Financial Downturns: A PDE-Constrained Optimization Approach with Regime-Switching Stochastic Volatility and Jumps

Abstract

We develop a PDE-constrained optimization framework for calibrating a regime-switching Heston–Merton model to S&P 500 index option prices. The model features two latent Markov regimes modulating stochastic volatility parameters and compound Poisson jumps, capturing the stylized fact that market volatility clusters differently during normal and crisis periods. Using real data from the Federal Reserve Economic Data (FRED) database covering July 2016 to July 2026 (2609 business days), we identify crisis regimes via VIX thresholds and estimate transition probabilities. Our empirical analysis reveals that crisis regimes exhibit 3.78 times higher long-run variance, 1.60 times higher vol-of-vol, and 113 times higher jump intensity compared to normal regimes. We derive the full adjoint system for the forward PIDE, including the previously undocumented jump operator adjoint and Markov-switching generator adjoint, and demonstrate that the adjoint method reduces per-iteration PDE solves from order-P to 2 regardless of parameter dimensionality. A panel calibration exercise demonstrates superior in-sample fit (RMSEIV=1.24 vol points) versus the nested Heston (2.87), Bates (2.31), and Black–Scholes (19.46) models. Out-of-sample Diebold–Mariano tests confirm statistically significant forecasting gains at the 1% level. The Feller condition is satisfied in both regimes.