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Good–deal bounds are price bounds for a financial portfolio which depends on an individual trader's preferences. Mathematically, if A {\displaystyle A} is a set of portfolios with future outcomes which are "acceptable" to the trader, then define the function ρ : L p → R {\displaystyle \rho :{\mathcal {L}}^{p}\to \mathbb {R} } by
where A T {\displaystyle A_{T}} is the set of final values for self-financing trading strategies. Then any price in the range , ρ ] {\displaystyle ,\rho ]} does not provide a good deal for this trader, and this range is called the "no good-deal price bounds."
If A = { Z ∈ L 0 : Z ≥ 0 P − a . s . } {\displaystyle A=\left\{Z\in {\mathcal {L}}^{0}:Z\geq 0\;\mathbb {P} -a.s.\right\}} then the good-deal price bounds are the no-arbitrage price bounds, and correspond to the subhedging and superhedging prices. The no-arbitrage bounds are the greatest extremes that good-deal bounds can take.
If A = { Z ∈ L 0 : E ] ≥ E ] } {\displaystyle A=\left\{Z\in {\mathcal {L}}^{0}:\mathbb {E} ]\geq \mathbb {E} ]\right\}} where u {\displaystyle u} is a utility function, then the good-deal price bounds correspond to the indifference price bounds.