We first review results on arbitrage theory for some notions of "simple" strategies, which do not allow continuous portfolio rebalancing by arbitrary amounts. Then, the focus of the talk is on trading under integer constraints, that is, we assume that the offered goods or shares are traded in integer quantities instead of the usual real quantity assumption. For finite probability spaces and rational asset prices this has little effect on the core of the theory of no-arbitrage pricing. For price processes not restricted to the rational numbers, a novel theory of integer arbitrage free pricing and hedging emerges. We establish an FTAP, involving a set of absolutely continuous martingale measures satisfying an additional property. The set of prices of a contingent claim is not necessarily an interval, but is either empty or dense in an interval. We also discuss superhedging with integral portfolios. Joint work with Paul Eisenberg.
Dynamic trading under integer constraints
Some Explicit Results on Dynkin Games with Incomplete and Asymmetric Information
In this talk I will consider two types of Dynkin game with non-standard information structures. The first one is a zero-sum game between two players who observe a geometric Brownian motion but in which the minimiser knows the drift of the process whereas the maximiser doesn't know it. We construct an explicit Nash equilibrium in which the uninformed player uses a pure strategy and the informed player uses a randomised strategy. The second game is a non-zero sum game between two agents interested in the purchase of the same asset. Neither of the two players knows with certainty whether their competitor is `active' and in that sense that they have uncertain competition. Also in this case we construct explicitly a Nash equilibrium in which both players randomise their strategy.