A comparison of techniques for dynamic risk measures with transaction costs (original) (raw)

2013, arXiv: Risk Management

This paper contains an overview of results for dynamic risk measures in markets with transaction costs. We provide the main results of four different approaches. We will prove under which assumptions results within these approaches coincide, and how properties like primal and dual representation and time consistency in the different approaches compare to each other.

Time consistency of dynamic risk measures in markets with transaction costs

2013

Set-valued dynamic risk measures are defined on L_d^p(\mathcal F_T) with 0 \leq p \leq \infty and with an image space in the power set of L_d^p(\mathcal F_t). Primal and dual representations of dynamic risk measures are deduced. Definitions of different time consistency properties in the set-valued framework are given. It is shown that the recursive form for multivariate risk measures as well as an additive property for the acceptance sets is equivalent to a stronger time consistency property called multi-portfolio time consistency.

Pricing under dynamic risk measures

Open Mathematics, 2019

In this paper, we study the discrete-time super-replication problem of contingent claims with respect to an acceptable terminal discounted cash flow. Based on the concept of Immediate Profit, i.e., a negative price which super-replicates the zero contingent claim, we establish a weak version of the fundamental theorem of asset pricing. Moreover, time consistency is discussed and we obtain a representation formula for the minimal super-hedging prices of bounded contingent claims.

A comparison of techniques for dynamic multivariate risk measures

2013

This paper contains an overview of results for dynamic multivariate risk measures. We provide the main results of four different approaches. We will prove under which assumptions results within these approaches coincide, and how properties like primal and dual representation and time consistency in the different approaches compare to each other.

Scalar multivariate risk measures with a single eligible asset

2020

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 w...

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