We show that a law-invariant pricing functional defined on a general Orlicz space is typically incompatible with frictionless risky assets in the sense that one and only one of the following alternatives can hold: Either every risky payoff has a strictly-positive bid-ask spread or the pricing functional is given by an expectation and, hence, every payoff has zero bid-ask spread. In doing so we extend and unify a variety of "collapse to the mean" results from the literature and highlight the key role played by law invariance in causing the collapse.As a byproduct, we derive a number of applications to law-invariant acceptance sets and risk measures as well as Schur-convex functionals .
We establish general versions of a variety of results for quasiconvex, lower-semicontinuous, and law-invariant functionals. Our results extend well-known results from the literature to a large class of spaces of random variables. We sometimes obtain sharper versions, even for the well-studied case of bounded random variables. Our approach builds on two fundamental structural results for law-invariant functionals: the equivalence of law invariance and Schur convexity, i.e., monotonicity with respect to the convex stochastic order, and the fact that a law-invariant functional is fully determined by its behaviour on bounded random variables. We show how to apply these results to provide a unifying perspective on the literature on law-invariant functionals, with special emphasis on quantile-based representations, including Kusuoka representations, dilatation monotonicity, and infimal convolutions .
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show that a
establish general versions of a variety of results for quasiconvex, lower-semicontinuous, and
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pricing functional defined on a general Orlicz space is typically incompatible with frictionless risky assets in the sense that one and only one of the following alternatives can hold: Either every risky payoff has a strictly-positive bid-ask spread or the pricing functional is given by an expectation and, hence, every payoff has zero bid-ask spread. In doing so we extend and unify a variety of "collapse to the mean"
functionals. Our results extend well-known
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and highlight the key role played by law invariance in causing the collapse.As a byproduct, we derive a number of applications to
to a large class of spaces of random variables. We sometimes obtain sharper versions, even for the well-studied case of bounded random variables. Our approach builds on two fundamental structural results for
R
acceptance sets and risk measures as well as Schur-convex functionals
functionals: the equivalence of law invariance and Schur convexity, i.e., monotonicity with respect to the convex stochastic order, and the fact that a law-invariant functional is fully determined by its behaviour on bounded random variables. We show how to apply these results to provide a unifying perspective on the literature on law-invariant functionals, with special emphasis on quantile-based representations, including Kusuoka representations, dilatation monotonicity, and infimal convolutions