Coherent Pricing

Recent literature proved the existence of an unbounded market price of risk (MPR) or maximum generalized Sharpe ratio (GSR) if one combines the most important Brownian-motion-linked arbitrage free pricing models with a coherent and expectation bounded risk measure. Furthermore, explicit sequences of portfolios with a theoretical (risk, return) diverging to (􀀀1;+1) were constructed and their performance tested. The empirical evidence revealed that the divergence to (􀀀1;+1) is only theoretical (not real), but the MPR is much larger than the GSR of the most important international stock indices. The natural question is how to modify the available pricing models so as to prevent the caveat above. The theoretical MPR cannot equal inf nity but must be large enough (consistent with the empirical findings) and this will be the focus of this paper. It will be shown that every arbitrage free pricing model can be improved in such a manner that the new stochastic discount factor (SDF) satisfie the two requirements above, and the newMPR becomes bounded but large enough. This is important for several reasons; Firstly, if the existent models predict unrealistic price evolutions then these mistakes may imply important capital losses to practitioners and theoretical errors to researchers. Secondly, the lack of an unbounded MPR is much more coherent and consistent with equilibrium. Finally, the major discrepancies between the initial pricing model and the modifie one will affect the tails of their SDF, which seems to justify several empirical caveats of previous literature. For instance, it has been pointed out that it is not easy to explain the real quotes of many deeply OTM options with the existing pricing models.