Peacocks and the Zeta distributions

We prove in this short paper that the stochastic process defined by: $$Y_{t} := \frac{X_{t+1}}{\mathbb{E}\left[ X_{t+1}\right]},\; t\geq a > 1,$$ is an increasing process for the convex order,where Χt a random variable taking values in N with probability P(Χt = n) = n-t/(𝛇(t)) and 𝛇(t) = +∞∑k=1(1/kt), ∀t > 1.

On the Increasing Convex Order of Relative Spacings of Order Statistics

Relative spacings are relative differences between order statistics. In this context, we extend previous results concerning the increasing convex order of relative spacings of two distributions from the case of consecutive spacings to general spacings. The sufficient conditions are given in terms of the expected proportional shortfall order. As an application, we compare relative deprivation within some parametric families of income distributions.

UNIVERSALLY MARKETABLE INSURANCE UNDER MULTIVARIATE MIXTURES

The study of desirable structural properties that define a marketable insurance contract has been a recurring theme in insurance economic theory and practice. In this article, we develop probabilistic and structural characterizations for insurance indemnities that are universally marketable in the sense that they appeal to all policyholders whose risk preferences respect the convex order. We begin with the univariate case where a given policyholder faces a single risk, then extend our results to the case where multiple risks possessing a certain dependence structure coexist. The non-decreasing and 1-Lipschitz condition, in various forms, is shown to be intimately related to the notion of universal marketability. As the highlight of this article, we propose a multivariate mixture model which not only accommodates a host of dependence structures commonly encountered in practice but is also flexible enough to house a rich class of marketable indemnity schedules.

Stochastic dominance relations for generalised parametric distributions obtained through composition

Investigating stochastic dominance within flexible multi-parametric families of distributions is often complicated, owing to the high number of parameters or non-closed functional forms. To simplify the problem, we use the T–X method, making it possible to obtain generalised models through the composition of cumulative distributions and quantile functions. We derive conditions for the second-order stochastic dominance and for the increasing convex order within multi-parametric families in two steps, namely: (i) breaking them down via the T–X approach and (ii) checking dominance conditions of the (more) manageable distributions composing the model. We apply our method to some special distributions and focus on the beta-generated family, which enables the comparisons of order statistics of i.i.d. samples from (possibly) different random variables.

Bounds of Adj-TVaR Prediction for Aggregate Risk

In financial and insurance industries, risks may come from several sources. It is therefore important to predict future risk by using the concept of aggregate risk. Risk measure prediction plays important role in allocating capital as well as in controlling (and avoiding) worse risk. In this paper, we consider several risk measures such as Value-at-Risk (VaR), Tail VaR (TVaR) and its extension namely Adjusted TVaR (Adj-TVaR). Specifically, we perform an upper bound for such risk measure applied for aggregate risk models. The concept and property of comonotonicity and convex order are utilized to obtain such upper bound.Keywords: Coherent property, comonotonic rv, convex order, tail property, Value-at-Risk (VaR).

A Black–Scholes inequality: applications and generalisations

The space of call price curves has a natural noncommutative semigroup structure with an involution. A basic example is the Black–Scholes call price surface, from which an interesting inequality for Black–Scholes implied volatility is derived. The binary operation is compatible with the convex order, and therefore a one-parameter sub-semigroup gives rise to an arbitrage-free market model. It is shown that each such one-parameter semigroup corresponds to a unique log-concave probability density, providing a family of tractable call price surface parametrisations in the spirit of the Gatheral–Jacquier SVI surface. An explicit example is given to illustrate the idea. The key observation is an isomorphism linking an initial call price curve to the lift zonoid of the terminal price of the underlying asset.