Acceptance set

In financial mathematics, acceptance set is a set of acceptable future net worth which is acceptable to the regulator. It is related to risk measures. In financial mathematics, acceptance set is a set of acceptable future net worth which is acceptable to the regulator. It is related to risk measures. Given a probability space ( Ω , F , P ) {displaystyle (Omega ,{mathcal {F}},mathbb {P} )} , and letting L p = L p ( Ω , F , P ) {displaystyle L^{p}=L^{p}(Omega ,{mathcal {F}},mathbb {P} )} be the Lp space in the scalar case and L d p = L d p ( Ω , F , P ) {displaystyle L_{d}^{p}=L_{d}^{p}(Omega ,{mathcal {F}},mathbb {P} )} in d-dimensions, then we can define acceptance sets as below. An acceptance set is a set A {displaystyle A} satisfying: An acceptance set (in a space with d {displaystyle d} assets) is a set A ⊆ L d p {displaystyle Asubseteq L_{d}^{p}} satisfying: Additionally, if A {displaystyle A} is convex (a convex cone) then it is called a convex (coherent) acceptance set. Note that K M = K ∩ M {displaystyle K_{M}=Kcap M} where K {displaystyle K} is a constant solvency cone and M {displaystyle M} is the set of portfolios of the m {displaystyle m} reference assets. An acceptance set is convex (coherent) if and only if the corresponding risk measure is convex (coherent). As defined below it can be shown that R A R ( X ) = R ( X ) {displaystyle R_{A_{R}}(X)=R(X)} and A R A = A {displaystyle A_{R_{A}}=A} . 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 The acceptance set associated with the entropic risk measure is the set of payoffs with positive expected utility. That is