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

$\rho(X) = \inf\left\{t \in \mathbb{R}: \exists V_T \in A_T: X + t + V_T \in A\right\} = \inf\left\{t \in \mathbb{R}: \exists V_T \in A_T: X + t \in A - A_T\right\}$

where A_T is the set of final values for self-financing trading strategies. Then any price in the range ( − ρ(X),ρ( − 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 (μ,ρ)-Portfolio Optimization. 
2. ^ ^{a b c} John R. Birge (2008). Financial Engineering. Elsevier. pp. 521–524. ISBN 9780444517814. 
3. ^ Arai, Takuji; Fukasawa, Masaaki (2011) (pdf). Convex risk measures for good deal bounds. http://arxiv.org/PS_cache/arxiv/pdf/1108/1108.1273v1.pdf. Retrieved October 14, 2011.