Acceptance set

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:

1. A \supseteq L^p_+
2. A \cap L^p_{--} = \emptyset such that L^p_{--} = {X \in L^p: \forall \omega \in \Omega, X(\omega)
3. A \cap L^p_- = {0}
4. Additionally if A is convex then it is a convex acceptance set
5. 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:

1. u \in K_M \Rightarrow u1 \in A with 1 denoting the random variable that is constantly 1 \mathbb{P}-a.s.
2. u \in -\mathrm{int}K_M \Rightarrow u1 \not\in A
3. A is directionally closed in M with A + u1 \subseteq A ; \forall u \in K_M
4. 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

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 : A = {-V_T: (V_t)_{t=0}^T \text{ is the price of a self-financing portfolio at each time}}.

Entropic risk measure

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 : 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

References

1. (1999). "Coherent Measures of Risk". Mathematical Finance.
2. (2010). "Duality for Set-Valued Measures of Risk". SIAM Journal on Financial Mathematics.
3. (2010). "Encyclopedia of Quantitative Finance".