Coherent and Convex Fair Pricing and Variability Measures

In this paper, we show that coherent upper and lower previsions as well as coherent risk measures are only meaningful under the assumption that one starts with initial wealth being constantly 0. This implies at least for coherent upper and lower previsions a correction of their interpretation, especially coherent upper previsions turn out to represent infimum short selling prices instead of infimum selling prices and coherent lower previsions represent fair prices. We elaborate this meaning of coherent lower previsions by identifying a class of coherent variability measures and present a way to extend coherence to all possible situations of initial wealth. Since a coherent risk measure is the negative of a coherent lower prevision, all results presented in this paper can easily be reformulated in terms of risk measures. Finally, we sketch how corresponding results can be obtained when replacing coherence by convexity.