Spectral XVA: Replacing Monte Carlo in Counterparty Credit Risk

We propose a spectral method for computing Credit Valuation Adjustment (CVA) and related XVA metrics that eliminates Monte Carlo simulation entirely. The Fokker--Planck generator of the interest rate process is discretized in cosine basis, producing a matrix \(M \in \mathbb{R}^{N \times N}\) from which the expected exposure profile at any future date is computed via matrix exponential: \(A(t) = e^{Mt}A(0)\). The expected positive exposure \(\mathbb{E}[\max(V(r_t), 0)]\) --- normally the computational bottleneck requiring nested simulation --- reduces to an inner product between the spectral density and the portfolio value function. On a 10-swap Vasicek portfolio with 40 quarterly monitoring dates, the spectral method produces CVA within 1.3\% of Monte Carlo (100,000 paths) while running 16\(\times\) faster (0.1s vs 1.2s), with zero sampling noise and instant stress testing. For credit spread scenarios, the spectral method reuses the exposure profile and only reweights the default probabilities, achieving 22\(\times\) speedup over MC which requires full resimulation. We project that for a realistic bank desk (5,000 counterparties, 20 regulatory stress scenarios), the spectral approach reduces the XVA computation from hours to minutes.