Paper: arXiv 2212.09957

Authors: Marc Chataigner, Areski Cousin, Stéphane Crépey, Matthew Dixon, Djibril Gueye

Abstract

We explore the abilities of two machine learning approaches for no-arbitrage interpolation of European vanilla option prices, which jointly yield the corresponding local volatility surface: a finite dimensional Gaussian process (GP) regression approach under no-arbitrage constraints based on prices, and a neural net (NN) approach with penalization of arbitrages based on implied volatilities. We demonstrate the performance of these approaches relative to the SSVI industry standard. The GP approach is proven arbitrage-free, whereas arbitrages are only penalized under the SSVI and NN approaches. The GP approach obtains the best out-of-sample calibration error and provides uncertainty quantification.The NN approach yields a smoother local volatility and a better backtesting performance, as its training criterion incorporates a local volatility regularization term.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a mathematically rigorous approach to shape-constrained Gaussian processes for guaranteed arbitrage-free local volatility surface construction, contrasting it with a penalized neural network approach. Its empirical evaluation includes backtesting and calibration performance against standard industry benchmarks like SSVI.

Research Flowchart

  flowchart TD
    A[Goal: No-Arbitrage Option Interpolation & Local Volatility Surface Extraction] --> B[Inputs: European Vanilla Option Market Prices]
    B --> C{Methodology & Models}
    
    C --> D[Constrained Gaussian Process Regression]
    C --> E[Neural Network with Arbitrage Penalization]
    C --> F[Baseline Benchmark: SSVI Model]
    
    D --> G[Outcomes & Performance Comparison]
    E --> G
    F --> G
    
    G --> H[Findings: GP guarantees no-arbitrage, lowest out-of-sample error, and UQ; NN yields smoother Local Vol & best backtesting via regularization]