Spectral Counterparty Credit Risk: Deterministic EPE, PFE, and Regulatory Capital Without Monte Carlo

We derive the full counterparty credit risk (CCR) metric stack --- Expected Positive Exposure (EPE), Potential Future Exposure (PFE), Effective Expected Positive Exposure (EEPE), and Exposure at Default (EAD) --- from the Fokker--Planck spectral generator without Monte Carlo simulation. The spectral density \(p(r, t) = \sum_k A_k(t)\,\varphi_k(r)\), evolved via matrix exponential \(A(t) = e^{Mt}A(0)\), provides the exact conditional distribution of the risk factor at each monitoring date. EPE is an inner product between density coefficients and payoff projections. PFE at the \(q\)-th quantile is obtained by inverting the spectral CDF --- a root-finding problem on a smooth function, yielding a deterministic, noise-free PFE that does not depend on the number of simulation paths. EEPE and EAD follow algebraically.

On the 10-swap Vasicek benchmark portfolio, we compute: (i) the full EPE profile in 221ms (vs 4.5s simple MC, 29.2s nested MC-on-MC in optimized Rust); (ii) PFE at the 97.5th percentile with zero sampling noise (MC PFE has \(\sim\)5\% relative noise at \(10^5\) paths); (iii) EAD under the Internal Model Method (IMM) and compare with the Standardized Approach for CCR (SA-CCR); and (iv) regulatory capital under Basel III CRE52. The spectral approach yields IMM-quality exposure metrics at a fraction of the computational cost, potentially enabling banks currently on SA-CCR to adopt model-based approaches without the infrastructure investment traditionally required.