Good–deal bounds

Good–deal bounds are price bounds for a financial portfolio which depends on an individual trader's preferences. Mathematically, if ${\displaystyle A}$ is a set of portfolios with future outcomes which are "acceptable" to the trader, then define the function ${\displaystyle \rho$ :{\mathcal {L}}^{p}\to \mathbb {R} } by

${\displaystyle \rho (X)=\inf \left\{t\in \mathbb {R}$ :\exists V_{T}\in A_{T}:X+t+V_{T}\in A\right\}=\inf \left\{t\in \mathbb {R} :X+t\in A-A_{T}\right\}}

where ${\displaystyle A_{T}}$ is the set of final values for self-financing trading strategies. Then any price in the range ${\displaystyle (-\rho (X),\rho (-X))}$ does not provide a good deal for this trader, and this range is called the "no good-deal price bounds."[1][2]

If ${\displaystyle A=\left\{Z\in {\mathcal {L}}^{0}:Z\geq 0\;\mathbb {P} -a.s.\right\}}$ 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.[2][3]

If ${\displaystyle A=\left\{Z\in {\mathcal {L}}^{0}:\mathbb {E} [u(Z)]\geq \mathbb {E} [u(0)]\right\}}$ where ${\displaystyle u}$ is a utility function, then the good-deal price bounds correspond to the indifference price bounds.[2]