Wishart conditional tail risk measures: An analytic approach

This study introduces a new analytical framework for quantifying multivariate risk measures. Using the Wishart process, which is a stochastic process with values in the space of positive definite matrices, we derive several conditional tail risk measures which, thanks to the remarkable analytical properties of the Wishart process, can be explicitly computed up to a one- or two-dimensional integration. These quantities can also be used to solve analytically a capital allocation problem based on conditional moments. Exploiting the stochastic differential equation property of the Wishart process, we show how an intertemporal (i.e., time-lagged) view of these risk measures can be embedded in the proposed framework. Several numerical examples show that the framework is versatile and operational, thus providing a useful tool for risk management.

💡 Research Summary

This paper introduces a novel analytical framework for computing multivariate conditional tail risk measures by exploiting the mathematical properties of the Wishart process, a stochastic process that takes values in the cone of positive‑definite matrices. Traditional approaches to tail risk—such as Tail Conditional Expectation (TCE), tail variance, and higher‑order conditional moments—typically rely on Monte‑Carlo simulation, copula constructions, or distribution‑specific closed forms that are limited to particular families (e.g., multivariate gamma or elliptical distributions). The authors overcome these limitations by grounding the entire methodology in the affine structure of the Wishart process, whose moment generating function (MGF) can be expressed in closed form via a matrix Riccati differential equation.

The paper proceeds as follows. Section 2 defines the Wishart process through the stochastic differential equation
dxₜ = (ω + m xₜ + xₜ mᵀ) dt + √xₜ dWₜ σ + σᵀ dWₜᵀ √xₜ,
where ω, σ ∈ S⁺⁺ₙ and m ∈ ℝⁿˣⁿ satisfy stability conditions (e.g., ω ≽ βσ² with β ≥ n + 1). The authors recall that the process is affine, which implies that the MGF Φ(t,θ) = E