Computing Volatility Surfaces using Generative Adversarial Networks with Minimal Arbitrage Violations

Paper: arXiv 2304.13128

Authors: Andrew Na, Meixin Zhang, Justin Wan

Abstract

In this paper, we propose a generative adversarial network (GAN) approach for efficiently computing volatility surfaces. The idea is to make use of the special GAN neural architecture so that on one hand, we can learn volatility surfaces from training data and on the other hand, enforce no-arbitrage conditions. In particular, the generator network is assisted in training by a discriminator that evaluates whether the generated volatility matches the target distribution. Meanwhile, our framework trains the GAN network to satisfy the no-arbitrage constraints by introducing penalties as regularization terms. The proposed GAN model allows the use of shallow networks which results in much less computational costs. In our experiments, we demonstrate the performance of the proposed method by comparing with the state-of-the-art methods for computing implied and local volatility surfaces. We show that our GAN model can outperform artificial neural network (ANN) approaches in terms of accuracy and computational time.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents an innovative application of GANs to volatility surface calibration while directly incorporating non-arbitrage soft constraints. It balances solid mathematical formulation of arbitrage penalties with empirical validation comparing speed and accuracy against standard ANN architectures.

Research Flowchart

  flowchart TD
    A[Research Goal: Compute Volatility Surfaces Efficiently with Minimal Arbitrage Violations] --> B[Data Inputs: Historical Options Data & Implied/Local Volatility Surfaces]
    B --> C[Methodology: Train Shallow GAN Architecture]
    C --> D[Discriminator Evaluates Generated Volatility vs Target Distribution]
    C --> E[Regularization: Apply No-Arbitrage Penalties During Training]
    D --> F[Model Output: Generated Implied and Local Volatility Surfaces]
    E --> F
    F --> G[Outcomes: Outperforms ANNs in Accuracy, Computational Speed, and Arbitrage Compliance]