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In financial mathematics, acceptance sets are a mathematical concept related to risk measures. As the name suggests an acceptance set is a set of acceptable future net worth, which are acceptable to the regulator.
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) < 0\}
- 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[1]
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. [2] 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.[3] References
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