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SPX-VIX Risk Computations Via Perturbed Optimal Transport

Charlie Che, Hanxuan Lin, Yudong Yang, Guofan Hu, Lei Fang

Abstract

We propose a model independent framework for generating SPX and VIX risk scenarios based on a joint optimal transport calibration of their market smiles. Starting from the entropic martingale optimal transport formulation of Guyon, we introduce a perturbation methodology that computes sensitivities of the calibrated coupling using a Fisher information linearization. This allows risk to be generated without performing a full recalibration after market shocks. We further introduce a dimension reduction method based on perturbed optimal transport that produces fast and stable risk estimates while preserving the structural properties of the calibrated model. The approach is combined with Skew Stickiness Ratio(SSR) dynamics to translate SPX shocks into perturbations of forward variance and VIX distributions. Numerical experiments show that the proposed method produces accurate risk estimates relative to full recalibration while being computationally much faster. A backtesting study also demonstrates improved hedging performance compared with stochastic local volatility models.

SPX-VIX Risk Computations Via Perturbed Optimal Transport

Abstract

We propose a model independent framework for generating SPX and VIX risk scenarios based on a joint optimal transport calibration of their market smiles. Starting from the entropic martingale optimal transport formulation of Guyon, we introduce a perturbation methodology that computes sensitivities of the calibrated coupling using a Fisher information linearization. This allows risk to be generated without performing a full recalibration after market shocks. We further introduce a dimension reduction method based on perturbed optimal transport that produces fast and stable risk estimates while preserving the structural properties of the calibrated model. The approach is combined with Skew Stickiness Ratio(SSR) dynamics to translate SPX shocks into perturbations of forward variance and VIX distributions. Numerical experiments show that the proposed method produces accurate risk estimates relative to full recalibration while being computationally much faster. A backtesting study also demonstrates improved hedging performance compared with stochastic local volatility models.
Paper Structure (70 sections, 12 theorems, 134 equations, 11 figures, 4 tables)

This paper contains 70 sections, 12 theorems, 134 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Main Contributions
  4. Market Consistency And Risk Propagation
  5. Martingality And Consistency Conditions
  6. Martingality condition
  7. SPX–VIX consistency condition
  8. Practical considerations
  9. Mathematical Framework For Entropic Projections
  10. Finite Discrete Setup
  11. Entropic Optimal Transport
  12. Gauge Fixing And Numerical Stability
  13. Log-partition function.
  14. Perturbation Theory: General Marginal Shocks
  15. Admissible Perturbations
  16. ...and 55 more sections

Key Result

Theorem 2.1

Assume $\bar{\mu}(x) > 0$ for all $x$. If $\mathcal{P}(\mu_1,\mu_V,\mu_2)$ is nonempty, then problem eq:primal admits a unique minimizer $\mu^\star$.

Figures (11)

  • Figure 1: Estimated VIX SSR term structure across four historical windows.
  • Figure 2: SPX–VIX basis time series for the 1-month tenor over a two-year window.
  • Figure 3: SPX smile fit quality (two expiries)
  • Figure 4: VIX smile fit: observed vs model-implied
  • Figure 5: SPX--VIX calibration theoretical conditions
  • ...and 6 more figures

Theorems & Definitions (26)

  • Theorem 2.1: Existence and Uniqueness
  • proof
  • Theorem 2.2: Primal–Dual Equivalence
  • proof
  • Theorem 3.1: Well-posedness of Entropic Projections
  • proof
  • Lemma 3.2: Strict concavity and smoothness of the dual
  • proof
  • Theorem 3.3: Differentiability of optimal potentials and coupling
  • proof
  • ...and 16 more