Essays On The Mathematics Of Market Efficiency
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In 1970, Fama defined an efficient market as one where prices always 'fully reflect' available information, but so far a rigorous definition has been lacking. This thesis addresses this issue by providing a definition based on economic equilibria. Efficiency is then characterized in terms of Merton's No Dominance condition together with absence of arbitrage in the sense of No Free Lunch With Vanishing Risk, as well as the existence of an equivalent (true) martingale measure for the discounted price process. The stability of the efficiency property with respect to changes in the information set is investigated. In particular, efficiency is preserved under information reduction, but not necessarily under information expansion. Next, checkable necessary and sufficient conditions for efficiency are provided for a large class of high dimensional stochastic volatility models. Finally, information reduction is studied in the inefficient setting. This leads to new results on filtration shrinkage for strict local martingales.