Time consistency for scalar multivariate risk measures

In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive formulation of multivariate scalar risk measures to multiportfolio time consistency. We are motivated to study time consistency of multivariate scalar risk measures as the superhedging risk measure in markets with transaction costs (with a single eligible asset) (Jouini and Kallal (1995), Löhne and Rudloff (2014), Roux and Zastawniak (2016)) does not satisfy the usual scalar concept of time consistency. In fact, as demonstrated in (Feinstein and Rudloff (2021)), scalar risk measures with the same scalarization weight at all times would not be time consistent in general. The deduced recursive relation for the scalarizations of multiportfolio time consistent set-valued risk measures provided in this paper requires consideration of the entire family of scalarizations. In this way we develop a direct notion of a “moving scalarization” for scalar time consistency that corroborates recent research on scalarizations of dynamic multi-objective problems (Karnam, Ma and Zhang (2017), Kováčová and Rudloff (2021)).

Bipolar behavior of submodular, law-invariant capacities

In the case of a submodular, law-invariant capacity, we provide an entirely elementary proof of a result of Marinacci [M. Marinacci, Upper probabilities and additivity, Sankhyā Ser. A 61 1999, no. 3, 358–361]. As a corollary, we also show that the anticore of a continuous submodular, law-invariant nonatomic capacity has a dichotomous nature: either it is one-dimensional or it is infinite-dimensional. The results have implications for the use of such capacities in financial and economic applications.

The functional kNN estimator of the conditional expectile: Uniform consistency in number of neighbors

The main purpose of the present paper is to investigate the problem of the nonparametric estimation of the expectile regression in which the response variable is scalar while the covariate is a random function. More precisely, an estimator is constructed by using the k Nearest Neighbor procedures (kNN). The main contribution of this study is the establishment of the Uniform consistency in Number of Neighbors (UNN) of the constructed estimator. The usefulness of our result for the smoothing parameter automatic selection is discussed. Short simulation results show that the finite sample performance of the proposed estimator is satisfactory in moderate sample sizes. We finally examine the implementation of this model in practice with a real data in financial risk analysis.

On the elicitability of range value at risk

The debate of which quantitative risk measure to choose in practice has mainly focused on the dichotomy between value at risk (VaR) and expected shortfall (ES). Range value at risk (RVaR) is a natural interpolation between VaR and ES, constituting a tradeoff between the sensitivity of ES and the robustness of VaR, turning it into a practically relevant risk measure on its own. Hence, there is a need to statistically assess, compare and rank the predictive performance of different RVaR models, tasks subsumed under the term “comparative backtesting” in finance. This is best done in terms of strictly consistent loss or scoring functions, i.e., functions which are minimized in expectation by the correct risk measure forecast. Much like ES, RVaR does not admit strictly consistent scoring functions, i.e., it is not elicitable. Mitigating this negative result, we show that a triplet of RVaR with two VaR-components is elicitable. We characterize all strictly consistent scoring functions for this triplet. Additional properties of these scoring functions are examined, including the diagnostic tool of Murphy diagrams. The results are illustrated with a simulation study, and we put our approach in perspective with respect to the classical approach of trimmed least squares regression.

Kernel estimation for Lévy driven stochastic convolutions

We consider a Lévy driven stochastic convolution, also called continuous time Lévy driven moving average model X ⁢ ( t ) = ∫ 0 t a ⁢ ( t - s ) ⁢ d Z ⁢ ( s ) X(t)=\int_{0}^{t}a(t-s)\,dZ(s) , where 𝑍 is a Lévy martingale and the kernel a ( . ) a(\,{.}\,) a deterministic function square integrable on R + \mathbb{R}^{+} . Given 𝑁 i.i.d. continuous time observations ( X i ⁢ ( t ) ) t ∈ [ 0 , T ] (X_{i}(t))_{t\in[0,T]} , i = 1 , … , N i=1,\dots,N , distributed like ( X ⁢ ( t ) ) t ∈ [ 0 , T ] (X(t))_{t\in[0,T]} , we propose two types of nonparametric projection estimators of a 2 a^{2} under different sets of assumptions. We bound the L 2 \mathbb{L}^{2} -risk of the estimators and propose a data driven procedure to select the dimension of the projection space, illustrated by a short simulation study.

On the extension property of dilatation monotone risk measures

Let 𝒳 be a subset of L 1 L^{1} that contains the space of simple random variables ℒ and ρ : X → ( - ∞ , ∞ ] \rho\colon\mathcal{X}\to(-\infty,\infty] a dilatation monotone functional with the Fatou property. In this note, we show that 𝜌 extends uniquely to a σ ⁢ ( L 1 , L ) \sigma(L^{1},\mathcal{L}) lower semicontinuous and dilatation monotone functional ρ ¯ : L 1 → ( - ∞ , ∞ ] \overline{\rho}\colon L^{1}\to(-\infty,\infty] . Moreover, ρ ¯ \overline{\rho} preserves monotonicity, (quasi)convexity and cash-additivity of 𝜌. We also study conditions under which ρ ¯ \overline{\rho} preserves finiteness on a larger domain. Our findings complement extension and continuity results for (quasi)convex law-invariant functionals. As an application of our results, we show that transformed norm risk measures on Orlicz hearts admit a natural extension to L 1 L^{1} that retains robust representations.

Continuous-time limits of multi-period cost-of-capital margins

We consider multi-period cost-of-capital valuation of a liability cash flow subject to repeated capital requirements that are partly financed by capital injections from capital providers with limited liability. Limited liability means that, in any given period, the capital provider is not liable for further payment in the event that the capital provided at the beginning of the period turns out to be insufficient to cover both the current-period payments and the updated value of the remaining cash flow. The liability cash flow is modeled as a continuous-time stochastic process on {[0,T]}. The multi-period structure is given by a partition of {[0,T]} into subintervals, and on the corresponding finite set of times, a discrete-time cost-of-capital-margin process is defined. Our main objective is the analysis of existence and properties of continuous-time limits of discrete-time cost-of-capital-margin processes corresponding to a sequence of partitions whose meshes tend to zero. Moreover, we provide explicit expressions for the limit processes when cash flows are given by Itô diffusions and processes with independent increments.