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Deep Learning-Enhanced Calibration of the Heston Model: A Unified Framework

Arman Zadgar, Somayeh Fallah, Farshid Mehrdoust

TL;DR

The paper addresses the computational bottleneck of calibrating the Heston stochastic volatility model by introducing a hybrid deep learning framework that cascades two feedforward networks. The Price Approximator Network (PAN) learns a fast surrogate for the option price surface as a function of strike/moneyness, while the Calibration Correction Network (CCN) refines Heston outputs to better match market prices, enabling real-time calibration. Empirical results on S&P 500 options demonstrate substantial reductions in pricing error metrics (e.g., RMSE, MAE, MRE) and faster convergence compared with traditional calibration, across both in-sample and out-of-sample tests, including SP 500 Mini options. This approach offers a practical, robust enhancement for real-time financial model calibration and can generalize to other stochastic-volatility frameworks demanding speed and precision.

Abstract

The Heston stochastic volatility model is a widely used tool in financial mathematics for pricing European options. However, its calibration remains computationally intensive and sensitive to local minima due to the model's nonlinear structure and high-dimensional parameter space. This paper introduces a hybrid deep learning-based framework that enhances both the computational efficiency and the accuracy of the calibration procedure. The proposed approach integrates two supervised feedforward neural networks: the Price Approximator Network (PAN), which approximates the option price surface based on strike and moneyness inputs, and the Calibration Correction Network (CCN), which refines the Heston model's output by correcting systematic pricing errors. Experimental results on real S\&P 500 option data demonstrate that the deep learning approach outperforms traditional calibration techniques across multiple error metrics, achieving faster convergence and superior generalization in both in-sample and out-of-sample settings. This framework offers a practical and robust solution for real-time financial model calibration.

Deep Learning-Enhanced Calibration of the Heston Model: A Unified Framework

TL;DR

The paper addresses the computational bottleneck of calibrating the Heston stochastic volatility model by introducing a hybrid deep learning framework that cascades two feedforward networks. The Price Approximator Network (PAN) learns a fast surrogate for the option price surface as a function of strike/moneyness, while the Calibration Correction Network (CCN) refines Heston outputs to better match market prices, enabling real-time calibration. Empirical results on S&P 500 options demonstrate substantial reductions in pricing error metrics (e.g., RMSE, MAE, MRE) and faster convergence compared with traditional calibration, across both in-sample and out-of-sample tests, including SP 500 Mini options. This approach offers a practical, robust enhancement for real-time financial model calibration and can generalize to other stochastic-volatility frameworks demanding speed and precision.

Abstract

The Heston stochastic volatility model is a widely used tool in financial mathematics for pricing European options. However, its calibration remains computationally intensive and sensitive to local minima due to the model's nonlinear structure and high-dimensional parameter space. This paper introduces a hybrid deep learning-based framework that enhances both the computational efficiency and the accuracy of the calibration procedure. The proposed approach integrates two supervised feedforward neural networks: the Price Approximator Network (PAN), which approximates the option price surface based on strike and moneyness inputs, and the Calibration Correction Network (CCN), which refines the Heston model's output by correcting systematic pricing errors. Experimental results on real S\&P 500 option data demonstrate that the deep learning approach outperforms traditional calibration techniques across multiple error metrics, achieving faster convergence and superior generalization in both in-sample and out-of-sample settings. This framework offers a practical and robust solution for real-time financial model calibration.

Paper Structure

This paper contains 24 sections, 35 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Financial model calibration
  3. Heston's Stochastic Volatility Model
  4. Calibration Framework
  5. Neural Networks in Model Calibration
  6. Neural Network Function Approximation
  7. Understanding Calibration Slowness
  8. Approximating the Pricing Function Using ANN
  9. Using the Trained Neural Network for Model Calibration
  10. Feedforward Neural Network Architecture
  11. Framework for Neural Network Modeling
  12. Training Strategies and Optimization Algorithms
  13. Weight Initialization and Kaiming Method
  14. Neural Network Training Mechanics
  15. Cost Function in Neural Network Training
  16. ...and 9 more sections

Figures (8)

  • Figure 1: Price Approximator Network (PAN) architecture.
  • Figure 2: Calibration Correction Network (CCN) architecture.
  • Figure 3: Results of the calibration of the Heston's option pricing model on $S\&P$ 500 option data.
  • Figure 4: Approximation of a line using the Price Approximator Network (PAN) model on $S\&P$ 500 option.
  • Figure 5: Results of improving the Heston's option pricing model calibration using deep learning for $S\&P$ 500 data.
  • ...and 3 more figures