Applying Deep Learning to Calibrate Stochastic Volatility Models

ArXiv ID: 2309.07843 “View on arXiv”

Authors: Unknown

Abstract

Stochastic volatility models, where the volatility is a stochastic process, can capture most of the essential stylized facts of implied volatility surfaces and give more realistic dynamics of the volatility smile/skew. However, they come with the significant issue that they take too long to calibrate. Alternative calibration methods based on Deep Learning (DL) techniques have been recently used to build fast and accurate solutions to the calibration problem. Huge and Savine developed a Differential Machine Learning (DML) approach, where Machine Learning models are trained on samples of not only features and labels but also differentials of labels to features. The present work aims to apply the DML technique to price vanilla European options (i.e. the calibration instruments), more specifically, puts when the underlying asset follows a Heston model and then calibrate the model on the trained network. DML allows for fast training and accurate pricing. The trained neural network dramatically reduces Heston calibration’s computation time. In this work, we also introduce different regularisation techniques, and we apply them notably in the case of the DML. We compare their performance in reducing overfitting and improving the generalisation error. The DML performance is also compared to the classical DL (without differentiation) one in the case of Feed-Forward Neural Networks. We show that the DML outperforms the DL. The complete code for our experiments is provided in the GitHub repository: https://github.com/asridi/DML-Calibration-Heston-Model

Keywords: stochastic volatility, Heston model, calibration, differential machine learning (DML), neural networks, Derivatives (Options)

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced stochastic calculus and differential equations for the Heston model, alongside complex deep learning and regularization techniques, indicating high mathematical density. It includes a public GitHub repository with implementation code, rigorous comparisons between Differential Machine Learning and classical methods, and focuses on calibration speed and accuracy for practical application, demonstrating strong empirical readiness.

  flowchart TD
    A["Research Goal"] --> B["Data Generation"]
    A["Research Goal"] --> C["Methodology"]
    B --> C
    C --> D["Computational Process"]
    D --> E["Findings"]

    subgraph A ["Research Goal"]
        A1["<b>Apply DML to Heston Calibration</b><br>Improve speed/accuracy of SV model calibration"]
    end

    subgraph B ["Data & Inputs"]
        B1["<b>Simulated Data</b><br>Features: Moneyness, Maturity<br>Labels: Option Price<br>Differentials: Vega, Delta"]
    end

    subgraph C ["Key Methodology"]
        C1["<b>Neural Network Training</b><br>1. Standard DL (MSE Loss)<br>2. DML (inc. diff. terms)"]
        C2["<b>Regularization</b><br>Applied to reduce overfitting"]
    end

    subgraph D ["Computational Process"]
        D1["<b>Optimization Loop</b><br>Calibrate Heston params (k, θ, ρ, σ, v0)<br>to match network predictions"]
    end

    subgraph E ["Key Findings"]
        E1["<b>DML Outperforms DL</b><br>Higher accuracy, lower error"]
        E2["<b>Significant Speedup</b><br>Drastic reduction in calibration time vs. traditional methods"]
    end