title: Good–deal bounds
text: # Good–deal bounds
  
  Good–deal bounds are price bounds for a financial portfolio which depends on an individual trader's preferences. Mathematically, if $A$ is a set of portfolios with future outcomes which are "acceptable" to the trader, then define the function $\rho :{\mathcal {L}}^{p}\to \mathbb {R}$ by
  
      $\rho (X)=\inf \left\{t\in \mathbb {R} :\exists V_{T}\in A_{T}:X+t+V_{T}\in A\right\}=\inf \left\{t\in \mathbb {R} :X+t\in A-A_{T}\right\}$
  
  where $A_{T}$ is the set of final values for self-financing trading strategies. Then any price in the range $(-\rho (X),\rho (-X))$ does not provide a good deal for this trader, and this range is called the "no good-deal price bounds."
  
  If $A=\left\{Z\in {\mathcal {L}}^{0}:Z\geq 0\;\mathbb {P} -a.s.\right\}$ 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 $A=\left\{Z\in {\mathcal {L}}^{0}:\mathbb {E} [u(Z)]\geq \mathbb {E} [u(0)]\right\}$ where $u$ is a utility function, then the good-deal price bounds correspond to the indifference price bounds.
  
  