Scalar Multivariate Risk Measures with a Single Eligible Asset

Zachary Feinstein, Birgit Rudloff

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

In this paper, we present results on scalar risk measures in markets with transaction costs. Such risk measures are defined as the minimal capital requirements in the cash asset. First, some results are provided on the dual representation of such risk measures, with particular emphasis given on the space of dual variables as (equivalent) martingale measures and prices consistent with the market model. Then, these dual representations are used to obtain the main results of this paper on time consistency for scalar risk measures in markets with frictions. It is well known from the superhedging risk measure in markets with transaction costs that the usual scalar concept of time consistency is too strong and not satisfied. We will show that a weaker notion of time consistency can be defined, which corresponds to the usual scalar time consistency but under any fixed consistent pricing process. We will prove the equivalence of this weaker notion of time consistency and a certain type of backward recursion with respect to the underlying risk measure with a fixed consistent pricing process. Several examples are given, with special emphasis on the superhedging risk measure.

Original languageEnglish
Pages (from-to)899-922
Number of pages24
JournalMathematics of Operations Research
Volume47
Issue number2
DOIs
StatePublished - May 2022

Keywords

  • dynamic risk measures
  • scalarization
  • time consistency

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