Rough Martingale Optimal Transport: Theory, Implementation, and Regulatory Applications for Non-Modelable Risk Factors

TL;DR

This paper addresses the challenge of pricing non-modelable risk factors under FRTB by introducing Rough Martingale Optimal Transport (RMOT), which regularizes the transport problem with a rough volatility prior to yield finite, explicit extrapolation bounds. It establishes identifiability and error bounds for rough volatility with sparse data, proves correlation identifiability in multi-asset settings via distinct Hurst exponents, and provides a scalable block-sparse Newton calibration algorithm applicable to portfolios of up to $N \approx 50$. Empirical validation on SPY/QQQ and live SPX/NDX data demonstrates stretched exponential tail decay in the optimal martingale measure and significant capital relief (about $\$880M$ per $1B$ exotic book) compared with classical MOT, while maintaining rigorous error quantification and stability via cross-validation. The framework is designed to integrate with the FRTB ASA workflow, including a proposed backtesting protocol, and points to future work in rough path extensions and dynamic dependence. Key mathematical features include the single-asset rough Heston dynamics, LDP tail behavior $I(k) \sim C_H k^{1-H}$, explicit extrapolation bounds $|P_{RMOT}-P_{true}|$, and a multi-asset rough martingale copula that enforces cross-asset dependence through a rough covariance functional, all backed by principled identifiability results and scalable computation.

Abstract

The Fundamental Review of the Trading Book (FRTB) poses a significant challenge for exotic derivatives pricing, particularly for non-modelable risk factors (NMRF) where sparse market data leads to infinite audit bounds under classical Martingale Optimal Transport (MOT). We propose a unified Rough Martingale Optimal Transport (RMOT) framework that regularizes the transport plan with a rough volatility prior, yielding finite, explicit, and asymptotically tight extrapolation bounds. We establish an identifiability theorem for rough volatility parameters under sparse data, proving that 50 strikes are sufficient to estimate the Hurst exponent within $\pm 0.05$. For the multi-asset case, we prove that the correlation matrix is locally identifiable from marginal option surfaces provided the Hurst exponents are distinct. Model calibration on SPY and QQQ options (2019--2024) confirms that the optimal martingale measure exhibits stretched exponential tail decay ($\sim\exp(-k^{1-H})$), consistent with rough volatility asymptotics, whereas classical MOT yields trivial bounds. We validate the framework on live SPX/NDX data and scale it to $N = 30$ assets using a block-sparse optimization algorithm. Empirical results show that RMOT provides approximately \$880M in capital relief per \$1B exotic book compared to classical methods, while maintaining conservative coverage confirmed by 100-seed cross-validation. This constitutes a pricing framework designed to align with FRTB principles for NMRFs with explicit error quantification.

Figures (15)

• Figure 1: Single-asset RMOT pipeline from market option data through rough Heston calibration, identifiability verification, regularized MOT bounds computation, to final risk outputs with explicit error quantification.
• Figure 2: Multi-asset RMOT architecture. Single-asset RMOT marginals (left) feed into a rough martingale copula (center) that enforces correlation $\rho_{ij}$ via the rough covariance functional $\Psi_{ij}$.
• Figure 3: Parameter identifiability under sparse data (Theorem \ref{['thm:identifiability']}). Left: Effective dimension $d_{\text{eff}}$ grows logarithmically with strikes $m$, reaching full parameter space at $d_{\text{eff}}=5$ when $m \geq 50$. Right: Standard error of Hurst exponent estimate $\hat{H}$ decays as $m^{-1/2}$, matching the Cramér-Rao bound. Approximately 50 strikes are required to achieve $\pm 0.05$ precision at 95% confidence.
• Figure 4: Hurst Exponent Recovery vs Strikes. The empirical error (solid line) follows the theoretical $m^{-1/2}$ decay rate (dashed), confirming Theorem 3.1.
• Figure 5: Deep OTM Extrapolation (All Assets). Extrapolation error decay for SPY, QQQ, IWM, and GLD. The theoretical bound (dashed) consistently envelopes the empirical error (solid), validating Theorem 3.2 across distinct roughness regimes.

Theorems & Definitions (25)

• Definition 2.1: Rough Heston Model
• Definition 2.2: Fisher Information
• Definition 2.3: RMOT Problem
• Proposition 2.1: Exponential Tilting
• Theorem 2.2: LDP for Rough Volatility
• Definition 2.4: Rough Covariance Functional
• Definition 2.5: Optimization
• Theorem 3.1: Identifiability under Sparse Data
• proof
• Theorem 3.2: Extrapolation Error