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.

• The paper presents a non-parametric framework using perturbed optimal transport to accurately price SPX–VIX derivatives.
• It integrates entropic martingale optimal transport, Fisher information geometry, and dimensional reduction to compute Greeks efficiently.
• Empirical validations reveal that the method outperforms traditional stochastic models by reducing hedging variance and computational cost.

SPX–VIX Risk Computations via Perturbed Optimal Transport: A Technical Analysis

Introduction and Motivation

The calibration and risk computation for joint SPX–VIX derivatives remain a persistent challenge due to the complex relationship between the S&P 500 (SPX) options market and the implied volatility index (VIX) derivatives. Traditional stochastic volatility models—Heston, Bergomi, rough volatility—demand restrictive structural assumptions that risk introducing model misspecification and can lead to mis-calibrated joint smiles. Recent progress in non-parametric calibration via martingale optimal transport (MOT), most notably entropic MOT, enables exact static fit to both SPX and VIX option marginals [guyon2020spxvix, Guyon_SSRN_3853237], but practical risk sensitivity computation remains problematic due to heavy computational cost and the lack of explicit formulas linking cross-asset shocks to joint Greeks.

This work delivers Perturbed Optimal Transport (POT) as a unified non-parametric framework for exact pricing and efficient risk generation in SPX–VIX derivatives, integrating model-free calibration, linear response (LR) sensitivity, dimenionality reduction (DR), and empirical VIX volatility dynamics via constraints on the Skew Stickiness Ratio (SSR).

Entropic Martingale MOT Calibration

Joint calibration utilizes entropic MOT to yield a coupling $\mu(s_1, v, s_2)$ over spot and VIX grids that satisfies:

• Exact matching of SPX and VIX marginal prices (smiles/futures).
• SPX risk-neutral martingality: $\mathbb{E}[S_2\,|\,S_1,V] = S_1$.
• VIX-forward variance linkage: $\mathbb{E}[L(S_2/S_1)\,|\,S_1,V] = V^2$.

Calibration relies on iterative Sinkhorn scaling for marginal constraints and Newton-Raphson steps for enforcing conditional financial constraints. The resulting $\mu^\star$ is a strictly positive exponential family distribution.

Figure 1: The model achieves high-fidelity calibration to the observed SPX $T_1$ smile, illustrated for one expiry.

Figure 2: Observed versus model-implied VIX smile, demonstrating precise marginal fit in the VIX option market.

The model consistently achieves tight static fit while relaxing SPX–VIX basis inconsistencies observed in actual market data.

Figure 3: The SPX–VIX basis for the 1-month tenor over two years highlights persistent discrepancies between SPX-implied and traded VIX futures.

Figure 4: Martingality check—calibrated $\mu^\star$ robustly enforces the imposed no-arbitrage martingale condition.

Perturbation and Fisher-Information-Based Risk Computation

The core technical contribution is explicit sensitivity calculation using information geometry: since the calibrated coupling forms a finite exponential family, shocks to the SPX or VIX marginals propagate linearly, to first order, through the inverse Fisher information ($H^{-1}$) of the base coupling. This enables fast and accurate evaluation of Greeks, replacing expensive bump-and-recalibrate approaches.

Given a payoff $G$, and an admissible marginal perturbation direction $h$, the linear response formula is

$$\Pi'(0) = \langle h, \psi \rangle,\quad \psi = H^{-1}g_G,$$

where $g_G$ is the covariance of $G$ with sufficient statistics.

This system incorporates all calibration and financial constraints, including SSR-driven VIX volatility dynamics and market basis (when present).

Incorporating Empirical Volatility Dynamics: Skew Stickiness Ratio (SSR)

To address the static nature of classical MOT, the paper introduces SSR—an empirical rule quantifying how the VIX smile skews in response to changes in the underlying (SPX or VIX future level). The SSR coefficient is estimated using historical windows, capturing cross-regime smile behavior. The SSR enters the calibration constraints as additional linear conditions, which are compatible with both the Fisher-based LR risk engine and the DR method.

Figure 5: Six-month window estimation of VIX SSR term structure, evidencing empirical support for SSR-based smile dynamics.

Dimensional Reduction: Fast Nonlinear Risk Updates

The DR method exploits the observation that conditional on $(S_1,V)$, the kernel $\kappa(s_2|s_1,v)$ changes little for small shocks; thus, the risk update reduces to an entropic optimal transport problem of much lower dimension (the $(S_1,V)$ marginal), substantially lowering computational costs. The full joint is then reconstructed by coupling the updated $(S_1,V)$ marginal with the frozen conditional kernel.

This yields nearly exact (not merely first-order) risk propagation for finite, realistic shocks, provided the conditional invariance holds approximately.

Empirical Validation: Recalibration Benchmark vs. Perturbative Methods

The proposed methodology is evaluated by benchmarking the accuracy of computed SPX sensitivity (delta, vega) for VIX derivatives. The results demonstrate that LR-POT and DR-POT sensitivities nearly coincide with those from full recalibration, even in the wings, while reducing compute time by over an order of magnitude.

Figure 6: SPX delta risk for VIX options from LR-POT versus full recalibration displays minimal deviation, confirming accuracy of the perturbative approach.

Figure 7: SPX vega risk across VIX option strikes, with LR-POT and recalib curves nearly coincident across the smile.

Hedging Backtesting: Portfolio-Level Impact

The practical efficiency of the risk engine is validated through a comprehensive backtest involving 50 randomized VIX option portfolios hedged using both the POT-derived Greeks and a benchmark stochastic volatility (SV) model. Portfolios are dynamically hedged using VIX futures or SPX instruments, with hedge ratios set by either method's calculated cross-Greeks.

Key results:

• The POT hedge consistently reduces the hedged P&L variance compared to the SV hedge for the majority (often all) of randomized portfolios.
• The variance reduction is most pronounced during turbulent regimes, as shown via rolling window analysis.

  Figure 8: POT hedge produces lower hedging variance than SV hedge for all 50 VIX option portfolios using VIX futures; bars left of zero indicate POT's advantage.

  

  Figure 9: POT risk engine yields more stable hedging P&L during volatile market episodes, as evidenced by a lower 20-day rolling standard deviation.

  

  Figure 10: Similar improvement is observed when hedging using SPX futures and SPX vanillas, with almost all portfolios seeing reduced risk using POT.

  

  Figure 11: The reduction in P&L volatility delivered by POT is robust across different hedging instrument bases, especially in stressed periods.

Implications and Outlook

The approach breaks from reliance on specification risk inherent to parametric stochastic volatility models, providing a flexible, model-free, and theoretically consistent method to compute joint Greeks and carry out risk attribution in equity–volatility cross-asset portfolios. By establishing connections between Fisher information geometry and market risk propagation, the framework generalizes to other settings involving multi-asset or path-dependent derivatives, and opens avenues for further research in:

• Empirical SSR calibration in other asset classes.
• Coupling with neural operator architectures for scalable high-fidelity MOT-based calibration.
• Extension to robust hedging and stress testing in the presence of structural market breaks.

Conclusion

This work defines a rigorous, computationally efficient, and model-independent pipeline for pricing and risk generation in SPX–VIX derivatives, combining martingale optimal transport calibration, information geometric perturbation theory, and empirical volatility smile dynamics via SSR constraints. Numerical experiments confirm the accuracy and outperformance of the method versus standard stochastic volatility models in both cross-Greek estimation and realized hedging variance. These results firmly establish perturbed optimal transport as an advanced tool for cross-asset risk management and hedging in modern equity volatility markets (2603.10857).