Good–deal bounds explained

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: l{L} ^{p} \to R by\rho(X) = inf \left\{t \in R : \exists V _{T} \in A _{T:} X + t + V _{T} \in A\right\} = inf \left\{t \in 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 l{L} ^{0:} Z \geq 0 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.^[3]

If

A = \left\{Z \in l{L} ^{0:} E [u(Z)] \geq E [u( 0 )]\right\} where u is a utility function, then the good-deal price bounds correspond to the indifference price bounds.

Notes and References

1. Coherent Risk Measures, Valuation Bounds, and (\mu,\rho )-Portfolio Optimization. Stefan. Jaschke. Uwe. Kuchler. 2000.
2. Book: Financial Engineering. John R. Birge. 2008. Elsevier. 521–524. 978-0-444-51781-4.
3. Convex risk measures for good deal bounds. Takuji. Arai. Masaaki. Fukasawa. 2011. q-fin.PR. 1108.1273v1.