ON THE ASYMPTOTIC SHAPE OF QUANTILE SURFACES

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Highly relevant for quantitative risk managers and portfolio optimizers. It provides theoretical guarantees for the convexity/concavity of VaR surfaces in log-normal portfolios, which is crucial for ensuring that optimization algorithms converge to global minima/maxima rather than getting stuck in local extrema. It also offers explicit asymptotic formulas for optimal allocation in tail regimes.

Implementing the exact Hessian formula (Theorem 1.1) requires numerical integration over hypersurfaces or Monte Carlo estimation of conditional expectations. The asymptotic approximations (Section 4) are simpler to implement as they reduce to closed-form shape functions, but require careful handling of tail parameters.

The paper provides rigorous mathematical proofs and explicit formulas for the Hessian and asymptotic shapes. However, it is a theoretical paper without code or empirical datasets provided in the extract. Reproduction requires implementing the derived analytical formulas.