Mean‐ portfolio selection and ‐arbitrage for coherent risk measures

Herdegen, Martin; Khan, Nazem

doi:10.1111/mafi.12333

Cited by 13 publications

(27 citation statements)

References 51 publications

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“…The literature on mean-ρ portfolio selection for positively homogeneous risk measures has been discussed in detail in [34], and we refer the interested reader there. Here, we only focus on the case that ρ fails to be positively homogeneous.…”

Section: Related Literaturementioning

confidence: 99%

“…Indeed, even if the mean-ρ problem (1) has a solution, there might still be ρ-arbitrage -in which case the mean-ρ problem (2) does not have a solution; cf. [34,Corollary 3.21]. Finally, neither [48] nor [42] consider (Q3) which we believe to be the most interesting and important question, in particular from the point of view of the regulator.…”

Section: Related Literaturementioning

confidence: 99%

“…A key observation is that ρ satisfies weak/strong sensitivity to large losses if and only if ρ ∞ does. If ρ has a dual representation, then so does ρ ∞ , and we can lift the results from [34] on the dual characterisation of ρ-arbitrage for coherent risk measures to the case of convex risk measures; see Theorem 4.8. We also provide a dual characterisation of strong ρ-arbitrage for convex risk measures in Theorem 4.6.…”

Section: Introductionmentioning

confidence: 99%

“…These phenomena are referred to as ρ-arbitrage (a generalisation of arbitrage of the first kind) and strong ρ-arbitrage (a generalisation of arbitrage of the second kind), respectively. Both concepts have been studied extensively in Herdegen and Khan [34] from a theoretical perspective; the practical relevance of ρ-arbitrage has been discussed by Armstrong and Brigo in [6] and [5]. In particular, [34,Theorem 3.23] implies that regulators cannot exclude (strong) ρ-arbitrage a priori when imposing a positively homogeneous (monotone and cash-invariant) risk measure -unless ρ is as conservative as the worst-case risk measure.…”

Section: Introductionmentioning

confidence: 99%

“…Both concepts have been studied extensively in Herdegen and Khan [34] from a theoretical perspective; the practical relevance of ρ-arbitrage has been discussed by Armstrong and Brigo in [6] and [5]. In particular, [34,Theorem 3.23] implies that regulators cannot exclude (strong) ρ-arbitrage a priori when imposing a positively homogeneous (monotone and cash-invariant) risk measure -unless ρ is as conservative as the worst-case risk measure. Since a worst-case approach to risk is infeasible in practise, this indicates that one should move beyond the class of positively homogeneous risk measures for effective risk constraints in the context of portfolio selection.…”

Section: Introductionmentioning

confidence: 99%

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Sensitivity to large losses and $ρ$-arbitrage for convex risk measures

Herdegen¹,

Khan²

2022

Preprint

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This paper studies mean-risk portfolio selection in a one-period financial market, where risk is quantified by a star-shaped risk measure ρ. We introduce two new axioms: weak and strong sensitivity to large losses. We show that the first axiom is key to ensure the existence of optimal portfolios and the second one is key to ensure the absence of ρ-arbitrage.This leads to a new class of risk measures that are suitable for portfolio selection. We show that ρ belongs to this class if and only if ρ is real-valued and the smallest positively homogeneous risk measure dominating ρ is the worst-case risk measure.We then specialise to the case that ρ is convex and admits a dual representation. We derive necessary and sufficient dual characterisations of (strong) ρ-arbitrage as well as the property that ρ is suitable for portfolio selection. Finally, we introduce the new risk measure of Loss Sensitive Expected Shortfall, which is similar to and not more complicated to compute than Expected Shortfall but suitable for portfolio selection -which Expected Shortfall is not.

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Section: Related Literaturementioning

confidence: 99%

Section: Related Literaturementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Sensitivity to large losses and $ρ$-arbitrage for convex risk measures

Herdegen¹,

Khan²

2022

Preprint

Self Cite

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Fundamental theorem of asset pricing with acceptable risk in markets with frictions

Arduca

Munari

2023

Finance Stoch

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We study the range of prices at which a rational agent should contemplate transacting a financial contract outside a given market. Trading is subject to nonproportional transaction costs and portfolio constraints, and full replication by way of market instruments is not always possible. Rationality is defined in terms of consistency with market prices and acceptable risk thresholds. We obtain a direct and a dual description of market-consistent prices with acceptable risk. The dual characterisation requires an appropriate extension of the classical fundamental theorem of asset pricing where the role of arbitrage opportunities is played by good deals, i.e., costless investment opportunities with acceptable risk–reward tradeoff. In particular, we highlight the importance of scalable good deals, i.e., investment opportunities that are good deals regardless of their volume.

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Lower semicontinuity of monotone functionals in the mixed topology on $C_{b}$

Nendel

2024

Finance Stoch

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The main result of this paper characterises the continuity from below of monotone functionals on the space $C_{b}$ C b of bounded continuous functions on an arbitrary Polish space as lower semicontinuity in the mixed topology. In this particular situation, the mixed topology coincides with the Mackey topology for the dual pair $(C_{b},\mathrm{ca})$ ( C b , ca ) , where $\mathrm{ca}$ ca denotes the space of all countably additive signed Borel measures of finite variation. Hence lower semicontinuity in the mixed topology is for convex monotone maps $C_{b}\to \mathbb{R}$ C b → R equivalent to a dual representation in terms of countably additive measures. Such representations are of fundamental importance in finance, e.g. in the context of risk measures and superhedging problems. Based on the main result, regularity properties of capacities and dual representations of Choquet integrals in terms of countably additive measures for 2-alternating capacities are studied. Moreover, a well-known characterisation of star-shaped risk measures on $L^{\infty }$ L ∞ is transferred to risk measures on $C_{b}$ C b . In a second step, the paper provides a characterisation of equicontinuity in the mixed topology for families of convex monotone maps. As a consequence, for every convex monotone map on $C_{b}$ C b taking values in a locally convex vector lattice, continuity in the mixed topology is equivalent to continuity on norm-bounded sets.

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Mean‐ portfolio selection and ‐arbitrage for coherent risk measures

Abstract: We revisit mean-risk portfolio selection in a one-period financial market where risk is quantified by a positively homogeneous risk measure 𝜌. We first show that under

Cited by 13 publications

References 51 publications

Sensitivity to large losses and $ρ$-arbitrage for convex risk measures

Sensitivity to large losses and $ρ$-arbitrage for convex risk measures

Fundamental theorem of asset pricing with acceptable risk in markets with frictions

Lower semicontinuity of monotone functionals in the mixed topology on $C_{b}$

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