Acceptance set

An acceptance set is a set ${\displaystyle A}$ satisfying:

1. ${\displaystyle A\supseteq L_{+}^{p}}$
2. ${\displaystyle A\cap L_{--}^{p}=\emptyset }$ such that ${\displaystyle L_{--}^{p}=\{X\in L^{p}:\forall \omega \in \Omega ,X(\omega )<0\}}$
3. ${\displaystyle A\cap L_{-}^{p}=\{0\}}$
4. Additionally if ${\displaystyle A}$ is convex then it is a convex acceptance set
   1. And if ${\displaystyle A}$ is a positively homogeneous cone then it is a coherent acceptance set[1]

An acceptance set (in a space with ${\displaystyle d}$ assets) is a set ${\displaystyle A\subseteq L_{d}^{p}}$ satisfying:

1. ${\displaystyle u\in K_{M}\Rightarrow u1\in A}$ with ${\displaystyle 1}$ denoting the random variable that is constantly 1 ${\displaystyle \mathbb {P} }$-a.s.
2. ${\displaystyle u\in -\mathrm {int} K_{M}\Rightarrow u1\not \in A}$
3. ${\displaystyle A}$ is directionally closed in ${\displaystyle M}$ with ${\displaystyle A+u1\subseteq A\;\forall u\in K_{M}}$
4. ${\displaystyle A+L_{d}^{p}(K)\subseteq A}$

Additionally, if ${\displaystyle A}$ is convex (a convex cone) then it is called a convex (coherent) acceptance set. [2]

Note that ${\displaystyle K_{M}=K\cap M}$ where ${\displaystyle K}$ is a constant solvency cone and ${\displaystyle M}$ is the set of portfolios of the ${\displaystyle m}$ reference assets.

• If ${\displaystyle \rho }$ is a (scalar) risk measure then ${\displaystyle A_{\rho }=\{X\in L^{p}:\rho (X)\leq 0\}}$ is an acceptance set.
• If ${\displaystyle R}$ is a set-valued risk measure then ${\displaystyle A_{R}=\{X\in L_{d}^{p}:0\in R(X)\}}$ is an acceptance set.

• If ${\displaystyle A}$ is an acceptance set (in 1-d) then ${\displaystyle \rho _{A}(X)=\inf\{u\in \mathbb {R} :X+u1\in A\}}$ defines a (scalar) risk measure.
• If ${\displaystyle A}$ is an acceptance set then ${\displaystyle R_{A}(X)=\{u\in M:X+u1\in A\}}$ is a set-valued risk measure.

The acceptance set associated with the superhedging price is the negative of the set of values of a self-financing portfolio at the terminal time. That is

${\displaystyle A=\{-V_{T}:(V_{t})_{t=0}^{T}{\text{ is the price of a self-financing portfolio at each time}}\}}$.

The acceptance set associated with the entropic risk measure is the set of payoffs with positive expected utility. That is

${\displaystyle A=\{X\in L^{p}({\mathcal {F}}):E[u(X)]\geq 0\}=\{X\in L^{p}({\mathcal {F}}):E\left[e^{-\theta X}\right]\leq 1\}}$

where ${\displaystyle u(X)}$ is the exponential utility function.[3]