Neural Calibration of the Heston-Hull-White Hybrid Model

This repository provides a high-performance framework for the calibration of a three-factor stochastic model, combining Heston's stochastic volatility with the Hull-White short-rate process. The system utilizes a Deep Neural Network (DNN) to solve the inverse problem of mapping observed European option price surfaces to the latent physical parameters of the underlying Stochastic Differential Equations (SDEs).

1. Theoretical Framework

The model simulates the joint evolution of the asset price $S_t$, its instantaneous variance $v_t$, and the risk-free short rate $r_t$:

Asset and Variance Dynamics (Heston)

$$dS_t = r_t S_t dt + \sqrt{v_t} S_t dW_t^S$$ $$dv_t = \kappa_v (\theta_v - v_t) dt + \xi_v \sqrt{v_t} dW_t^v$$

Short Rate Dynamics (Hull-White)

$$dr_t = \kappa_r (\theta_r - r_t) dt + \xi_r dW_t^r$$

The correlation structure is defined by $d\langle W^S, W^v \rangle_t = \rho dt$. To ensure the variance process remains strictly positive, the Feller condition ($2\kappa_v \theta_v > \xi_v^2$) is enforced during synthetic data generation.

2. Computational Architecture

High-Performance Parallelization

The system employs a parallelized Monte Carlo engine optimized for high-core count environments (50 cores). It utilizes Python's native multiprocessing to generate $O(10^4)$ synthetic market scenarios. Each scenario represents a unique volatility surface spanning multiple strikes and tenors.

Memory-Efficient Pipeline

Data generation implements a chunked, disk-backed caching strategy using compressed .npz archives. Normalization is performed in-place to minimize RAM residency, allowing for large-scale training on restricted-memory systems.

3. Results and Validation

Parameter Recovery and Inversion

The model's ability to invert the SDE is validated through high-density calibration plots. By mapping normalized prices back to the physical parameters, the network achieves high $R^2$ scores across the test manifold.

Figure 1: High-density scatter plots of True vs. Predicted parameters. The convergence of points along the identity line demonstrates successful neural inversion.

Volatility Topology and Risk Sensitivity

The engine produces a full 3D Implied Volatility Surface, allowing for the analysis of the volatility smile and term structure. Furthermore, the system computes the Vega manifold ($\frac{\partial C}{\partial \sigma}$) to evaluate the financial sensitivity of the calibration.

Figure 2: (Left) Calibrated 3D IV Surface. (Center) Vega Risk Manifold. (Right) Risk-weighted error heatmap, quantifying the impact of residuals in basis points (bps).