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.

Mathematical Definition

Given a probability space, and letting be the Lp space in the scalar case and in d-dimensions, then we can define acceptance sets as below.

Scalar Case

An acceptance set is a set A satisfying:

Set-valued Case

An acceptance set (in a space with d assets) is a set satisfying: Additionally, if A is convex (a convex cone) then it is called a convex (coherent) acceptance set. Note that where K is a constant solvency cone and M is the set of portfolios of the m reference assets.

Relation to Risk Measures

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 and A_{R_A} = A.

Risk Measure to Acceptance Set

Acceptance Set to Risk Measure

Examples

Superhedging price

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

Entropic risk measure

The acceptance set associated with the entropic risk measure is the set of payoffs with positive expected utility. That is where u(X) is the exponential utility function.