TY - JOUR
T1 - Scalar Multivariate Risk Measures with a Single Eligible Asset
AU - Feinstein, Zachary
AU - Rudloff, Birgit
N1 - Publisher Copyright:
Copyright: © 2021 INFORMS.
PY - 2022/5
Y1 - 2022/5
N2 - 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.
AB - 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.
KW - dynamic risk measures
KW - scalarization
KW - time consistency
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U2 - 10.1287/moor.2021.1153
DO - 10.1287/moor.2021.1153
M3 - Article
AN - SCOPUS:85134255912
SN - 0364-765X
VL - 47
SP - 899
EP - 922
JO - Mathematics of Operations Research
JF - Mathematics of Operations Research
IS - 2
ER -