Mathematisches Kolloquium

Zusammenfassung: The classical result due to Delbaen and Schachermayer says that for a frictionless financial market with a semimartingale price process the NFLVR property (No Free Lunch with Vanishing Risk) holds if and only if there is an equivalent σ-martingale measure. On the other hand, the NFLVR property holds if and only if the NA property (No Arbitrage) holds jointly with the NAA1 property (the absence of asymptotic arbitrage opportunities of the first kind). Recently, Takaoka proved that NAA1 is equivalent to the existence of a local martingale deflator, that is, a strictly positive local martingale multiplicator transforming all positive value processes into local martingales. There is an important question whether the martingale deflator is the inverse of the value process corresponding to some portfolio (the latter is called in this case the market portfolio). It happens that in general the answer is negative though for the processes with the Levy measure concentrated in a finite number of points NAA1 implies the existence of market portfolio. We shall discuss also the Kardaras theorem: for a model with one risky asset the NAA1 property implies that in any neighborhood of the objective probability there is an equivalent probability measure with respect to which there exists a market portfolio.

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