Acceptance set

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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[edit]

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[edit]

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) < 0\}
  3.  A \cap L^p_- = \{0\}
  4. Additionally if A is convex then it is a convex acceptance set
    1. And if A is a positively homogeneous cone then it is a coherent acceptance set[1]

Set-valued Case[edit]

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. [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[edit]

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.[citation needed]

Risk Measure to Acceptance Set[edit]

  • 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[edit]

  • 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[edit]

Superhedging price[edit]

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[edit]

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.[3]

References[edit]

  1. ^ Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (1999). "Coherent Measures of Risk". Mathematical Finance 9 (3): 203–228. doi:10.1111/1467-9965.00068. 
  2. ^ Hamel, A. H.; Heyde, F. (2010). "Duality for Set-Valued Measures of Risk" (PDF). SIAM Journal on Financial Mathematics 1 (1): 66–95. doi:10.1137/080743494. Retrieved August 17, 2012.  edit
  3. ^ Follmer, Hans; Schied, Alexander (October 8, 2008). "Convex and Coherent Risk Measures" (PDF). Retrieved July 22, 2010.