Deterministic Log-CF Evaluation for Correlated Lognormal Sums

The moment-based Latent representation of correlated lognormal sums (Nagy, 2026, The Exact Latent Distribution of Correlated Lognormal Sums) relies on scaled moments c_k = m_k/k! whose logarithms grow quadratically in k in non-degenerate positive-weight cases. This creates an extreme dynamic range in the Padé Toeplitz systems used by that implementation. The reported experiments become impractical for \sigma_{\max} > 0.8, but this paper does not prove a general condition-number asymptotic, an impossibility result, or an information-theoretic lower bound. The Smooth Latent Operator (Nagy, 2026, The Smooth Latent Operator: Parameter-Free Distributional Representations via Kernel Moment Recovery) mitigates the observed instability through regularization without eliminating it in the reported high-volatility cases. We show that the moment-based route expands an unsuitable formal generating function. The moments are the formal Taylor coefficients associated with M(z)=E[e^{zS}], which is infinite for positive real z in the non-degenerate lognormal setting and has zero Taylor radius at the origin. By contrast, the Hermite-chaos expansion — the Wiener polynomial chaos of S in the underlying Gaussian variables — uses coefficients that decay factorially. This occurs because the exponential function e^{Y} has a convergent Hermite series for Y \sim N(\mu, \sigma^2), with coefficients \sigma^k/k!. The resulting Hermite Latent \Lambda^H = \{c_\mathbf{k}^H\} lives in \ell^2 (no Gaussian weight is needed). The characteristic function is represented by a convergent inner product (no Padé resummation needed). The CDF can then be approximated by Fourier-cosine inversion (Fang and Oosterlee, 2008). The numerical conditioning and accuracy of the full evaluator are assessed here only in the reported test regimes. We interpret this as a grade-3 Latent: a representation choice guided by the problem's structure. For lognormal sums, the Hermite-chaos basis is a natural candidate because it matches the underlying Gaussian variables. This is a problem-specific motivation, not a universal basis-selection theorem. The exact CDF integral representation is determined by the finite generative latent (w, \mu, \Sigma) \in \mathbb{R}^{n(n+5)/2} for positive weights and positive-definite covariance. Its numerical evaluator uses Gauss-Hermite order, COS order, and a finite log-domain as explicit precision choices. The reported results motivate this route as a useful alternative to moment-based evaluation; rigorous parameter-selection and end-to-end convergence guarantees remain open.

Maturity: Draft. Target venue: Mathematical Finance / Annals of Applied Probability. Part of The Latent research program.

Related papers in this program: Rough Volatility, Universal.