论文标题
均值 - $ρ$投资组合选择和$ρ$ -Arbitrage用于连贯的风险措施
Mean-$ρ$ portfolio selection and $ρ$-arbitrage for coherent risk measures
论文作者
论文摘要
我们在一个周期金融市场中重新访问平均风险投资组合选择，在该市场中，风险通过正质量的风险度量$ρ$量化。我们首先表明，在温和的假设下，固定回报的最佳投资组合集是非空的和紧凑的。但是，与经典的均值差异投资组合选择不同，可能发生不存在有效的投资组合。我们将这种情况称为$ρ$ - arbitrage，并证明它不能被排除 - 除非$ρ$与最坏的风险措施一样保守。 在提供了$ρ$ - arbitrage的原始特征之后，我们将注意力集中在接纳双重表示的连贯风险度量上，并给出$ρ$ - arbitrage的必要双重表征。我们表明，缺少$ρ$ - 贵族与折扣风险资产的同等标准措施（EMMS）之间的相互作用密切相关，并且在$ρ$的双重表示中，绝对连续的措施集。我们结果的一个特殊情况表明，市场不接收$ρ$ -ARBITRAGE的预期不足$α$时，并且只有存在EMM $ \ MATHBB {q} \ abor Mathbb {q Mathbb {p} $，以便$ \ vert \ frac {\ text {d} \ mathbb {q}}} {\ text {d} \ mathbb {p}} \ vert_ \ vert_ \ infty <\ infty <\ frac {1}α$。
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 mild assumptions, the set of optimal portfolios for a fixed return is nonempty and compact. However, unlike in classical mean-variance portfolio selection, it can happen that no efficient portfolios exist. We call this situation $ρ$-arbitrage, and prove that it cannot be excluded -- unless $ρ$ is as conservative as the worst-case risk measure. After providing a primal characterisation of $ρ$-arbitrage, we focus our attention on coherent risk measures that admit a dual representation and give a necessary and sufficient dual characterisation of $ρ$-arbitrage. We show that the absence of $ρ$-arbitrage is intimately linked to the interplay between the set of equivalent martingale measures (EMMs) for the discounted risky assets and the set of absolutely continuous measures in the dual representation of $ρ$. A special case of our result shows that the market does not admit $ρ$-arbitrage for Expected Shortfall at level $α$ if and only if there exists an EMM $\mathbb{Q} \approx \mathbb{P}$ such that $\Vert \frac{\text{d}\mathbb{Q}}{\text{d}\mathbb{P}} \Vert_\infty < \frac{1}α$.
