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 $\rho :{ℒ}^p\to ℝ$ by

$\rho (X)=\inf \{t\in ℝ:\mathrm{\exists }{V}_T\in A_T:X+t+V_T\in A\}=\inf \{t\in ℝ:X+t\in A-A_T\}$

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."

If $A=\{Z\in {ℒ}^0:Z\ge 0\mathbb{P}-a.s.\}$ 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.

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

References

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