Deep Learning Volatility

Abstract: We present a neural network based calibration method that performs the calibration task within a few milliseconds for the full implied volatility surface. The framework is consistently applicable throughout a range of volatility models -including the rough volatility family- and a range of derivative contracts. The aim of neural networks in this work is an off-line approximation of complex pricing functions, which are difficult to represent or time-consuming to evaluate by other means. We highlight how this perspective opens new horizons for quantitative modelling: The calibration bottleneck posed by a slow pricing of derivative contracts is lifted. This brings several numerical pricers and model families (such as rough volatility models) within the scope of applicability in industry practice. The form in which information from available data is extracted and stored influences network performance: This approach is inspired by representing the implied volatility and option prices as a collection of pixels. In a number of applications we demonstrate the prowess of this modelling approach regarding accuracy, speed, robustness and generality and also its potentials towards model recognition.

Deep Learning Volatility: A Neural Network Approach in Rough Volatility Models

This paper presents a novel calibration method for volatility models using deep learning techniques, specifically targeting rough volatility models, which are renowned for their realism but lack of tractability in industrial applications. The authors propose a framework that employs neural networks as high-dimensional functional approximators to accelerate the calibration process and circumvent existing computational bottlenecks in derivative pricing.

Main Contributions

The major contributions of this work are as follows:

1. Neural Network-Based Calibration: The paper focuses on a neural network approach to calibrate implied volatility surfaces in a few milliseconds across a range of stochastic models, including rough volatility models, which generally require costly Monte Carlo simulations for evaluation.
2. Off-Line Approximation: The framework hinges upon off-line approximation of complex pricing functions. Neural networks are trained to learn these functions, enabling the real-time generation of option prices once the network is calibrated to a specific model and market parameters.
3. Representation as Pixel Grid: The research proposes representing implied volatility surfaces as collections of pixels, drawing parallels to image-based data processing, which enhances accuracy, speed, and generality in model recognition and pricing tasks.
4. Validation through Numerical Experiments: The approach is validated through extensive numerical experiments, showcasing the speed, robustness, and accuracy of neural network-driven calibration across simulated and historical data.

Numerical and Qualitative Results

The neural network approach demonstrates considerable numerical performance, as evidenced by:

• Accuracy: The accuracy of neural network-based option price approximation remains within the error bounds of Monte Carlo simulations, with discrepancies primarily below 1% in implied volatility terms.
• Speed: Evaluations indicate a speed-up factor of up to 16,000 times over traditional Monte Carlo approaches, significantly enhancing the feasibility of real-time applications in trading and risk management.
• General Applicability: The network architecture and training setup show good generalization in unseen parameter domains, providing confidence in the model's predictability across various market conditions.

Implications and Future Directions

The adoption of neural networks in this capacity offers several implications for both theoretical development and practical applications:

• Industrial Relevance: The framework holds promise for widespread adoption in generating actionable financial derivatives prices in milliseconds, addressing a critical need for speed and accuracy in financial markets.
• Theoretical Exploration: From a theoretical standpoint, these methodologies open pathways to explore model translations between various stochastic processes, potentially enriching existing mathematical finance toolkits.
• Machine Learning in Quantitative Finance: Continued advancements in neural network design and computing power will bolster the application of machine learning in quantitative finance, yielding more innovative solutions to long-standing challenges like the calibration of complex financial instruments.

This paper offers a compelling case for integrating deep learning with financial modeling, emphasizing practical benefits while maintaining rigorous controls on numerical accuracy and computational efficiency. The findings suggest potential avenues for enriching volatility modeling techniques and encourage further exploration in combining AI technologies with financial theory.