Acceptance set
≈ 3 min read
Names in other languages
No names in other languages are held for this entry yet.
Article bodies are English and are not machine-translated. A name in another language is not a translation of the article.
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.
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 (\Omega,\mathcal{F},\mathbb{P}) , and letting L^p = L^p(\Omega,\mathcal{F},\mathbb{P}) be the Lp space in the scalar case and L_d^p = L_d^p(\Omega,\mathcal{F},\mathbb{P}) in d-dimensions, then we can define acceptance sets as below.
Scalar Case
An acceptance set is a set A satisfying: A \supseteq L^p_+ A \cap L^p_{--} = \emptyset such that L^p_{--} = \{X \in L^p: \forall \omega \in \Omega, X(\omega) A \cap L^p_- = \{0\} Additionally if A is convex then it is a convex acceptance set And if A is a positively homogeneous cone then it is a coherent acceptance set
Set-valued Case
An acceptance set (in a space with d assets) is a set A \subseteq L^p_d satisfying: u \in K_M \Rightarrow u1 \in A with 1 denoting the random variable that is constantly 1 \mathbb{P} -a.s. u \in -\mathrm{int}K_M \Rightarrow u1 \not\in A A is directionally closed in M with A + u1 \subseteq A \; \forall u \in K_M A + L^p_d(K) \subseteq A Additionally, if A is convex (a convex cone) then it is called a convex (coherent) acceptance set. Note that K_M = K \cap M where K is a constant solvency cone and M is the set of portfolios of the m 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 R_{A_R}(X) = R(X) and A_{R_A} = A .
Risk Measure to Acceptance Set
If \rho is a (scalar) risk measure then A_{\rho} = \{X \in L^p: \rho(X) \leq 0\} is an acceptance set. If R is a set-valued risk measure then A_R = \{X \in L^p_d: 0 \in R(X)\} is an acceptance set.
Acceptance Set to Risk Measure
If A is an acceptance set (in 1-d) then \rho_A(X) = \inf\{u \in \mathbb{R}: X + u1 \in A\} defines a (scalar) risk measure. If A is an acceptance set then R_A(X) = \{u \in M: X + u1 \in A\} 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 A = \{-V_T: (V_t)_{t=0}^T \text{ is the price of a self-financing portfolio at each time}\} .
Entropic risk measure
The acceptance set associated with the entropic risk measure is the set of payoffs with positive expected utility. That is 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\} where u(X) is the exponential utility function.
References
Text derived from https://en.wikipedia.org/w/index.php?oldid=1376831949 under CC BY-SA 4.0; modifications have been made.