arXiv subjects

Gusti van Zyl

Publications and source records attributed to Gusti van Zyl.

5 recordsLinked to original sources

arXiv2026

Distributionally robust Expected Shortfall for convex payoffs

We study distributionally robust Expected Shortfall when the distribution of the underlying is perturbed by a size quantified with optimal transport distance based on the quadratic cost function. In the dual version of the robust expectation problem, which is part of the robest expected shortfall problem, the computation of the so-called $λc$-transform $f^{λc}$ of payoff $f$ is required. We show that under the quadratic cost function there exists a tractable representation of $f^{λc}$, if $f$ is convex. Furthermore, we show that robust expected shortfall can be characterized as the solution of a 2-dimensional minimization problem. We apply these results to obtain a closed-form formula for robust, with respect to the risk-neutral distribution, Expected Shortfall of an unhedged call option, from the point of view of the writer.

math.PR↗

arXiv2023

A minimax approach to duality for linear distributional sensitivity testing

We consider the problem of finding the maximum of $\mathbb{E}_ν[f(X)]$ where $ν$ is allowed to vary over all the probability measures on a Polish space $S$ for which $d_c(μ,ν)\leq θ$, in which $d_c$ is an optimal transport distance, $f$ a real-valued function on $S$ satisfying some regularity, $μ$ a ``baseline" measure and $θ\geq 0$. Whereas some of the derivations of the dual version of this optimization problem rely on Fenchel duality, we impose compactness on $S$ to allow us to instead use K. Fan's minimax theorem, which does not require vector space structure. This allows one to avoid the use of vector spaces of measures, or dual variables other than the Lagrange multiplier.

math.PR↗

arXiv2019

A measure approximation theorem for Wasserstein-robust expected values

We consider the problem of finding the infimum, over probability measures being in a ball defined by Wasserstein distance, of the expected value of a bounded Lipschitz random variable on $\mathbf{R}^d$. We show that if the $σ-$algebra is approximated in by a sequence of $σ$-algebras in a certain natural sense, then the solutions of the induced approximated minimization problems converge to that of the initial minimization problem.

math.PR↗

arXiv2018

A framework for simulating systemic risk and its application to the South African banking sector

We present a network-based framework for simulating systemic risk that considers shock propagation in banking systems. In particular, the framework allows the modeller to reflect a top-down framework where a shock to one bank in the system affects the solvency and liquidity position of other banks, through systemic market risks and consequential liquidity strains. We illustrate the framework with an application using South African bank balance sheet data. Spikes in simulated assessments of systemic risk agree closely with spikes in documented subjective assessments of this risk. This indicates that network models can be useful for monitoring systemic risk levels. The model results are sensitive to liquidity risk and market sentiment and therefore the related parameters are important considerations when using a network approach to systemic risk modelling.

q-fin.RM↗

arXiv2009

Self-Referential Definition of Orthogonality

There has for longer been an interest in finding equivalent conditions which define inner product spaces, and the respective literature is considerable, see for instance Amir, which lists 350 such results. Here, in this tradition, an alternative definition of orthogonality is presented which does not make use of any inner product. This definition, in the spirit of the recently developed non-wellfounded set theory, is self-referential, or circulatory.

math.GM↗