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.

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 satisfying:
  2. such that
  4. Additionally if is convex then it is a convex acceptance set
  5. # And if is a positively homogeneous cone then it is a coherent [risk measure|coherent] acceptance set

Set-valued Case

An acceptance set is a set satisfying:
  1. with denoting the random variable that is constantly 1 -a.s.
  3. is directionally closed in with
Additionally, if is convex then it is called a convex acceptance set.
Note that where is a constant solvency cone and is the set of portfolios of the reference assets.

Relation to Risk Measures

An acceptance set is convex if and only if the corresponding risk measure is convex. As defined below it can be shown that and.

Risk Measure to Acceptance Set

  • If is a risk measure then is an acceptance set.
  • If is a set-valued risk measure then is an acceptance set.

Acceptance Set to Risk Measure

  • If is an acceptance set then defines a risk measure.
  • If is an acceptance set then is a set-valued 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 is the exponential utility function.