Given a probability space , and letting be the Lp space in the scalar case and in d-dimensions, then we can define acceptance sets as below.
An acceptance set is a set satisfying:
- such that
- Additionally if is convex then it is a convex acceptance set
An acceptance set (in a space with assets) is a set satisfying:
Note that where is a constant solvency cone and is the set of portfolios of the 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 and .
Risk Measure to Acceptance Set
- If is a (scalar) risk measure then is an acceptance set.
- If is a set-valued risk measure then is an acceptance set.
Acceptance Set to Risk Measure
- If is an acceptance set (in 1-d) then defines a (scalar) risk measure.
- If is an acceptance set then is a set-valued risk measure.
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
Entropic risk measure
The acceptance set associated with the entropic risk measure is the set of payoffs with positive expected utility. That is
- 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.
- 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.
- Follmer, Hans; Schied, Alexander (October 8, 2008). "Convex and Coherent Risk Measures" (PDF). Retrieved July 22, 2010.