“Acceptance set”的意思、由来-开放百科全书

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Acceptance set

释义

Mathematical Definition
Scalar Case Set-valued Case
Relation to Risk Measures
Risk Measure to Acceptance Set Acceptance Set to Risk Measure
Examples
Superhedging price Entropic risk measure
References

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:

such that
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^&91;1&93;

Set-valued Case

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

with denoting the random variable that is constantly 1 -a.s.
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

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|date=February 2011}}

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.

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.^[3]

References

1. ^{{cite journal|last=Artzner|first=Philippe|last2=Delbaen|first2=Freddy|last3=Eber|first3=Jean-Marc|last4=Heath|first4=David|year=1999|title=Coherent Measures of Risk|journal=Mathematical Finance|volume=9|issue=3|pages=203–228|doi=10.1111/1467-9965.00068}}
2. ^{{Cite journal | last1 = Hamel | first1 = A. H. | last2 = Heyde | first2 = F. | doi = 10.1137/080743494 | title = Duality for Set-Valued Measures of Risk | journal = SIAM Journal on Financial Mathematics | volume = 1 | issue = 1 | pages = 66–95 | year = 2010 | pmid = | pmc = | citeseerx = 10.1.1.514.8477 }}
3. ^{{Cite journal|last=Follmer|first=Hans|last2=Schied|first2=Alexander|date=October 8, 2008|title=Convex and Coherent Risk Measures|url=http://wws.mathematik.hu-berlin.de/~foellmer/papers/CCRM.pdf|accessdate=July 22, 2010}}

1 : Financial risk modeling

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