Quantitative Finance

An elementary representation route for sums of correlated lognormals; published with DOI and stated for review.

Derivative pricing has a clear mathematical target: compute the discounted risk-neutral value of a payoff under a specified model. For European vanilla contracts this target is often analytic or nearly analytic.

The collapse of a market's effective dimensionality during financial crises is an **endogenous phase transition** rather than a smooth response to exogenous shocks. Specifically, it is a discontinuous, history-dependent jump.

How many parameters does it take to represent a smooth probability density on a bounded domain to accuracy $\varepsilon$? We prove that for densities extending holomorphically to the Bernstein ellipse with parameter $\rho > 1$, the answer is $N = \Th

Pricing European basket options requires the distribution of a weighted sum of correlated lognormals — the Fenton Distribution (Nagy, 2026a) — which has no closed form.

Expected Shortfall backtesting under Basel III/IV suffers from an unmeasured structural weakness: Monte Carlo estimation of ES injects computational noise into the Acerbi-Székely (2014) test statistic, but the magnitude of this contamination has not

We present the Eigen-COS method, a deterministic algorithm that computes exact Value-at-Risk, closed-form Expected Shortfall, and the full CDF/PDF for weighted sums of correlated lognormal assets — without Monte Carlo simulation.

We present a deterministic, semi-analytical framework for computing the complete distribution of a portfolio's terminal value at horizon $T$ for correlated lognormal assets. Unlike traditional approaches, this method requires no Monte Carlo simulation.

Every formula in quantitative finance — CAPM, Markowitz, VaR, Sharpe ratio, GARCH — takes returns as input. Yet the standard definitions of return fail when prices cross zero: log-returns are undefined, and simple returns produce sign errors.

We develop a variance reduction framework for simulating rare events in correlated portfolios by exploiting the eigenvalue decomposition of the correlation matrix. The central observation is that the eigenvalue modes $Z_k$ — projections of the asset vector onto the eigenvectors of the correlation matrix — are mutually independent.

We prove that every Itô diffusion $dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t$ admits a complete, finite-dimensional representation through its Fokker--Planck generator discretized in cosine basis.

The CDF of a weighted sum of correlated lognormal random variables has lacked a tractable characterization since Fenton (1960). We show that eigenvalue conditioning of the correlation matrix, followed by Fourier-cosine inversion, yields an analytic, grid-free $N$-term spectral representation of that CDF: the **Spectral Lognormal Distribution**.