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."^[1][2]

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.^[2][3]

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.^[2]

References

1. Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ($\mu,\rho$)-Portfolio Optimization". delete character in |title= at position 48 (help)
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.