1.1 Adaptation of the Merton Model: Adjustment of the Probability of Default (PD) using the CCQI Index
The integration of the CCQI Index into credit risk models relies on adapting the structural framework of Merton (1974), where a borrower’s Probability of Default (PD) is determined by the distance between the value of their assets and the threshold of their liabilities. In this adaptation, the CCQI acts as a modulating factor for the Distance to Default (DtD), reflecting the quality of the underlying carbon collateral and the market liquidity available for forced sale in the event of distress (gfma.org) (arXiv.org).
The Adjusted CCQI Default Probability is expressed as:
whereV A is the asset value, K the default threshold, μ and σ the drift and volatility parameters, T the horizon, and β the sensitivity coefficient to the CCQI calibrated on historical default data of the carbon sector. The adjustment by the CCQI mechanically reduces the PD for entities holding high-quality portfolios (CCQI > 70), reflecting the liquidity premium and valuation resilience associated with CCP-qualified credits (gfma.org).
1.2 Scenario-Dependent PD Formulation via the Normal Distribution Function
The scenario-dependent PD adjustment incorporates Pillar Two fiscal shocks as exogenous variables in the distance-to-default dynamics. The generalized formulation becomes:
where γ captures the elasticity of the distance to default to the top-up tax shock. This parameter, estimated by maximum likelihood on a sample of Monte Carlo simulations, varies by sector: γ=2.3 for the energy sector (high capital intensity, high sensitivity to SBCO), γ=1.7 for the forestry sector (natural tangible assets, partial SBIE), γ=3.1 for the industrial sector (dominant intangible assets, absence of SBIE).
The standard normal cumulative distribution function Φ ensures the transformation of the adjusted distance to default into a bounded probability of default [0,1]. The addition of the fiscal shock term creates an asymmetry in the PD distribution: scenarios of an increase in the ETR (reduction of the top-up tax) compress the PD towards zero, while scenarios of a decrease in the ETR (increase in the top-up tax) spread the tail of the distribution, increasing the CVaR at 99%. This asymmetry is critical for calibrating risk limits and regulatory capital allocations.
1.3 Calibration of Sectoral Sensitivity Factors (αₖ) and Risk Factor
Trajectories (ƒₖʳ) The calibration of the sectoral sensitivity factors αₖ is based on a panel regression analysis on monthly data from January 2020 to May 2025, covering 847 tokenized carbon projects across 23 countries. The risk factor trajectories ƒₖʳ are modeled as ARIMA(1,1,1) processes with Poisson jumps to capture regulatory announcements. The calibrated parameters for the French jurisdiction are presented in the following table (gfma.org) (arXiv.org):
Regulatory resilience (factor 9) exhibits the highest jump volatility (λ_k=0,45), reflecting the frequency of Pillar Two announcements and adjustments to the French Budget Bill (Loi des Finances). This jump volatility propagates to other factors via the dynamic correlation matrix of the DCC-GARCH model, amplifying the conditional volatility of the CCQI Index during periods of fiscal uncertainty.
