Paper: arXiv 2406.02319

Abstract

We consider the path-dependent volatility (PDV) model of Guyon and Lekeufack (2023), where the instantaneous volatility is a linear combination of a weighted sum of past returns and the square root of a weighted sum of past squared returns. We discuss the influence of an additional parameter that unlocks enough volatility on the upside to reproduce the implied volatility smiles of S&P 500 and VIX options. This PDV model, motivated by empirical studies, comes with computational challenges, especially in relation to VIX options pricing and calibration. We propose an accurate \emph{“pathwise”} neural network approximation of the VIX which leverages on the Markovianity of the 4-factor version of the model. The VIX is learned pathwise as a function of the Markovian factors and the model parameters. We use this approximation to tackle the joint calibration of S&P 500 and VIX options, quickly sample VIX paths, and price derivatives that jointly depend on S&P 500 and VIX. As an interesting aside, we also show that this \emph{“time-homogeneous”}, low-parametric, Markovian PDV model is able to fit the whole surface of S&P 500 implied volatilities remarkably well.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper involves advanced stochastic calculus, path-dependent volatility specifications, and novel neural network approximations for VIX pricing, indicating high mathematical complexity; it directly addresses joint calibration to SPX and VIX options with empirical backtesting and computational implementation, showing strong empirical rigor.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Accurate PDV Model Pricing & Calibration"] --> B["Data & Model Inputs"]
  B --> C["Methodology<br>Markovian 4-Factor PDV Model"]
  C --> D["Computational Process<br>Pathwise Neural Network VIX Approx"]
  D --> E["Joint Calibration<br>S&P 500 & VIX Options"]
  E --> F["Key Findings & Outcomes"]
  
  B -->|S&P 500 & VIX Options Data| D
  B -->|Model Parameters| C
  
  F -->|Accurate VIX Pricing| G1["VIX Derivatives Pricing"]
  F -->|Fast Calibration| G2["Efficient Joint Calibration"]
  F -->|Surface Fit| G3["S&P 500 Implied Volatility Smile"]
  F -->|Markovian Structure| G4["Time-Homogeneous Model"]