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 <math>(\Omega,\mathcal{F},\mathbb{P})</math>, and letting <math>L^p = L^p(\Omega,\mathcal{F},\mathbb{P})</math> be the Lp space in the scalar case and <math>L_d^p = L_d^p(\Omega,\mathcal{F},\mathbb{P})</math> in d-dimensions, then we can define acceptance sets as below.
===Scalar Case=== An acceptance set is a set <math>A</math> satisfying: # <math>A \supseteq L^p_+</math> # <math>A \cap L^p_{--} = \emptyset</math> such that <math>L^p_{--} = \{X \in L^p: \forall \omega \in \Omega, X(\omega) < 0\}</math> # <math> A \cap L^p_- = \{0\}</math> # Additionally if <math>A</math> is convex then it is a convex acceptance set ## And if <math>A</math> is a positively homogeneous cone then it is a coherent acceptance set<ref>{{cite journal|last1=Artzner|first1=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|s2cid=6770585 }}</ref>
===Set-valued Case=== An acceptance set (in a space with <math>d</math> assets) is a set <math>A \subseteq L^p_d</math> satisfying: # <math>u \in K_M \Rightarrow u1 \in A</math> with <math>1</math> denoting the random variable that is constantly 1 <math>\mathbb{P}</math>-a.s. # <math> u \in -\mathrm{int}K_M \Rightarrow u1 \not\in A</math> # <math>A</math> is directionally closed in <math>M</math> with <math>A + u1 \subseteq A \; \forall u \in K_M</math> # <math>A + L^p_d(K) \subseteq A</math> Additionally, if <math>A</math> is convex (a convex cone) then it is called a '''convex (coherent) acceptance set'''. <ref>{{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 | citeseerx = 10.1.1.514.8477 }}</ref>
Note that <math>K_M = K \cap M</math> where <math>K</math> is a constant solvency cone and <math>M</math> is the set of portfolios of the <math>m</math> 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 <math>R_{A_R}(X) = R(X)</math> and <math>A_{R_A} = A</math>.{{citation needed|date=February 2011}}
===Risk Measure to Acceptance Set=== * If <math>\rho</math> is a (scalar) risk measure then <math>A_{\rho} = \{X \in L^p: \rho(X) \leq 0\}</math> is an acceptance set. * If <math>R</math> is a set-valued risk measure then <math>A_R = \{X \in L^p_d: 0 \in R(X)\}</math> is an acceptance set.
===Acceptance Set to Risk Measure=== * If <math>A</math> is an acceptance set (in 1-d) then <math>\rho_A(X) = \inf\{u \in \mathbb{R}: X + u1 \in A\}</math> defines a (scalar) risk measure. * If <math>A</math> is an acceptance set then <math>R_A(X) = \{u \in M: X + u1 \in A\}</math> is a set-valued risk measure.
==Examples== ===Superhedging price=== {{main|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 : <math>A = \{-V_T: (V_t)_{t=0}^T \text{ is the price of a self-financing portfolio at each time}\}</math>.
===Entropic risk measure=== {{main|Entropic risk measure}} The acceptance set associated with the entropic risk measure is the set of payoffs with positive expected utility. That is : <math>A = \{X \in L^p(\mathcal{F}): E[u(X)] \geq 0\} = \{X \in L^p(\mathcal{F}): E\left[e^{-\theta X}\right] \leq 1\}</math> where <math>u(X)</math> is the exponential utility function.<ref name="FS10">{{Cite book|last1=Follmer|first1=Hans|last2=Schied|first2=Alexander|date=2010|title=Encyclopedia of Quantitative Finance |pages=355–363 |chapter=Convex and Coherent Risk Measures |chapter-url=http://www.math.hu-berlin.de/~foellmer/papers/CCRM.pdf}}</ref>
==References== {{Reflist}}
Category:Financial risk modeling