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 ${\displaystyle \rho :{\mathcal {L}}^{p}\to \mathbb {R} }$ 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 $A_{T}$ 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 $u$ is a utility function, then the good-deal price bounds correspond to the indifference price bounds.^[2]

References

1. Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ($\mu ,\rho$)-Portfolio Optimization".
2. John R. Birge (2008). Financial Engineering. Elsevier. pp. 521–524. ISBN 978-0-444-51781-4.
3. Arai, Takuji; Fukasawa, Masaaki (2011). "Convex risk measures for good deal bounds" (pdf). Retrieved October 14, 2011.