Abstract
In an L ∞-framework, we present majorant-preserving and sandwich-preserving extension theorems for linear operators. These results are then applied to price systems derived by a reasonable restriction of the class of applicable equivalent martingale measures. Our results prove the existence of a no-good-deal pricing measure for price systems consistent with bounds on the Sharpe ratio. We treat both discrete- and continuous-time market models. Within this study we present definitions of no-good-deal pricing measures that are equivalent to the existing ones and extend them to discrete-time models. We introduce the corresponding version of dynamic no-good-deal pricing measures in the continuous-time setting.
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Notes
We recall that even though the σ-algebra \(\mathcal{F}_{t}\) is (up to P-null sets) countably generated, the space \(L^{\infty}(\mathcal{F}_{t})\) is not separable in general. For example, if Ω is the space of continuous functions C([0,t];ℝd) equipped with the sup-norm, then Ω is a separable topological space, and its Borel σ-algebra \(\mathcal{F}_{t}\) is countably generated, but the corresponding Banach space \(L^{\infty}(\mathcal{F}_{t})\) equipped with the ess-sup-norm is not a separable topological space. Indeed, if Ω is a Polish space, \(\mathcal{F}_{t}\) its Borel σ-algebra, and \(L^{\infty}(\mathcal{F}_{t})\) is separable, then Ω is at most countable.
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Acknowledgements
This research was mainly carried through during the visit of G. Di Nunno at Ecole Polytechnique with the support of Chair of Financial Risks of the Risk Foundation, Paris, and the visit of J. Bion-Nadal at University of Oslo with the support of CMA—Centre of Mathematics for Applications.
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Bion-Nadal, J., Di Nunno, G. Dynamic no-good-deal pricing measures and extension theorems for linear operators on L ∞ . Finance Stoch 17, 587–613 (2013). https://doi.org/10.1007/s00780-012-0195-y
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DOI: https://doi.org/10.1007/s00780-012-0195-y
Keywords
- Price operator
- Dynamic risk measure
- Extension theorem
- Representation theorem
- Fundamental theorem
- Equivalent martingale measure
- Good deal