Finance:Good–deal bounds
Good–deal bounds are price bounds for a financial portfolio which depends on an individual trader's preferences. Mathematically, if [math]\displaystyle{ A }[/math] is a set of portfolios with future outcomes which are "acceptable" to the trader, then define the function [math]\displaystyle{ \rho: \mathcal{L}^p \to \mathbb{R} }[/math] by
- [math]\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\} }[/math]
where [math]\displaystyle{ A_T }[/math] is the set of final values for self-financing trading strategies. Then any price in the range [math]\displaystyle{ (-\rho(X), \rho(-X)) }[/math] does not provide a good deal for this trader, and this range is called the "no good-deal price bounds."[1][2]
If [math]\displaystyle{ A = \left\{Z \in \mathcal{L}^0: Z \geq 0 \; \mathbb{P}-a.s.\right\} }[/math] 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 [math]\displaystyle{ A = \left\{Z \in \mathcal{L}^0: \mathbb{E}[u(Z)] \geq \mathbb{E}[u(0)]\right\} }[/math] where [math]\displaystyle{ u }[/math] is a utility function, then the good-deal price bounds correspond to the indifference price bounds.[2]
References
- ↑ Jaschke, Stefan; Kuchler, Uwe (2000). Coherent Risk Measures, Valuation Bounds, and ([math]\displaystyle{ \mu,\rho }[/math])-Portfolio Optimization.
- ↑ 2.0 2.1 2.2 John R. Birge (2008). Financial Engineering. Elsevier. pp. 521–524. ISBN 978-0-444-51781-4.
- ↑ Arai, Takuji; Fukasawa, Masaaki (2011). "Convex risk measures for good deal bounds". arXiv:1108.1273v1 [q-fin.PR].
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