In financial mathematics, acceptance sets are a mathematical concept related to risk measure

Risk measure

A Risk measure is used to determine the amount of an asset or set of assets to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator...

s. As the name suggests an acceptance set is a set of acceptable future net worth, which are acceptable to the regulator

Financial regulation

Financial regulation is a form of regulation or supervision, which subjects financial institutions to certain requirements, restrictions and guidelines, aiming to maintain the integrity of the financial system...

.

Mathematical Definition

Given a probability space , and letting be the Lp space

Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

 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:

1. 
2. such that
3. 
4. Additionally if is convex
   Convex set
   In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
   
    then it is a convex acceptance set
   1. And if is a positively homogeneous
      Homogeneous function
      In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
      
       cone then it is a coherent
      Coherent risk measure
      In the field of financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have...
      
       acceptance set

Set-valued Case

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. 
3. is directionally closed in with
4. 

Additionally, if is convex (a convex cone

Convex cone

In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.-Definition:...

) then it is called a convex (coherent) acceptance set.

Note that where is a constant solvency cone

Solvency cone

The solvency cone is a concept used in financial mathematics which models the possible trades in the financial market. This is of particular interest to markets with transaction costs...

 and is the set of portfolios of the 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 .

Risk Measure to Acceptance Set

• 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

• 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.

Superhedging price

The acceptance set associated with the superhedging price is the negative of the set of values of a self-financing portfolio

Self-financing portfolio

Self-financing portfolio, an important concept in financial mathematics.A portfolio is self-financing if there is no exogenous infusion or withdrawal of money; the purchase of a new asset must be financed by the sale of an old one.- Mathematical definition :...

 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

Utility

In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....

. That is

where is the exponential utility

Exponential utility

In economics exponential discounting is a specific form of the discount function, used in the analysis of choice over time...

function.