Acceptance set

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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[edit]

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[edit]

An acceptance set is a set satisfying:

  1. such that
  2. Additionally if is convex then it is a convex acceptance set
    1. And if is a positively homogeneous cone then it is a coherent acceptance set[1]

Set-valued Case[edit]

An acceptance set (in a space with assets) is a set satisfying:

  1. with denoting the random variable that is constantly 1 -a.s.
  2. is directionally closed in with

Additionally, if is convex (a convex cone) then it is called a convex (coherent) acceptance set. [2]

Note that where is a constant solvency cone and is the set of portfolios of the reference assets.

Relation to Risk Measures[edit]

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 .[citation needed]

Risk Measure to Acceptance Set[edit]

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

Acceptance Set to Risk Measure[edit]

  • If is an acceptance set (in 1-d) then defines a (scalar) risk measure.
  • If is an acceptance set then is a set-valued risk measure.

Examples[edit]

Superhedging price[edit]

Main article: 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[edit]

Main article: 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.[3]

References[edit]

  1. ^ Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (1999). "Coherent Measures of Risk". Mathematical Finance. 9 (3): 203–228. doi:10.1111/1467-9965.00068. 
  2. ^ Hamel, A. H.; Heyde, F. (2010). "Duality for Set-Valued Measures of Risk" (pdf). SIAM Journal on Financial Mathematics. 1 (1): 66–95. doi:10.1137/080743494. Retrieved August 17, 2012. 
  3. ^ Follmer, Hans; Schied, Alexander (October 8, 2008). "Convex and Coherent Risk Measures" (pdf). Retrieved July 22, 2010.