Dynamic no-good-deal pricing measures and extension theorems for linear operators on L ∞
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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.
KeywordsPrice operator Dynamic risk measure Extension theorem Representation theorem Fundamental theorem Equivalent martingale measure Good deal
Mathematics Subject Classification (2010)46E30 91B70
JEL ClassificationG12 G13
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.
- 2.Barrieu, P., El Karoui, N.: Pricing, hedging and optimally designing derivatives via minimization of risk measures. In: Carmona, R. (ed.) Volume on Indifference Pricing, pp. 77–147. Princeton University Press, Princeton (2009) Google Scholar
- 7.Bion-Nadal, J.: Dynamic pricing models calibrated on both liquid and illiquid assets. Int. J. Econ. Res. 9, 263–283 (2012) Google Scholar
- 8.Bion-Nadal, J., Kervarec, M.: Dynamic risk measuring under model uncertainty: Taking advantage of the hidden probability measure. Preprint ArXiv 1012.5850 (Dec. 2010)
- 21.Neveu, J.: Discrete Parameter Martingales. North Holland, Amsterdam (1974) Google Scholar
- 22.Peng, S.: Nonlinear expectations, nonlinear evaluations and risk measures. In: Back, K., Bielecki, T.R., Hipp, C., Peng, S., Schachermayer, W., Frittelli, M., Runggaldier, W.J. (eds.) Stochastic Methods in Finance. Lecture Notes in Mathematics, vol. 1856, pp. 165–253. Springer, Berlin (2004) CrossRefGoogle Scholar