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 transaction costs . It is well known from the typical example, the superhedging risk measure in markets with transaction costs (expressed in the cash asset) as in Jouini and Kallal (1995), Roux and Zastawniak (2016), and Loehne and Rudloff (2014), that the usual scalar concept of time consistency is too strong and not satisfied. More generally, when it comes to scalarizing a multivariate risk measure with the same scalarization weight at each time (as this is the case in this paper as the risk measures are expressed in the cash asset), one cannot expect these scalar risk measures to be time consistent. This aligns with recent research on scalarizations of multivariate problems as in Karnam et al. (2017) and Kovacova and Rudloff (2018). Surprisingly, 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.
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 , as in Jouini and Kallal (1995), Roux and Zastawniak (2016), and Loehne and Rudloff (2014), 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.
R
transaction costs
frictions
D
typical example, the
R
(expressed in the cash asset)
,
R
More generally, when it comes to scalarizing a multivariate risk measure with the same scalarization weight at each time (as this is the case in this paper as the risk measures are expressed in the cash asset), one cannot expect these scalar risk measures to be time consistent. This aligns with recent research on scalarizations of multivariate problems as in Karnam et al. (2017) and Kovacova and Rudloff (2018). Surprisingly, we
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