On martingale measures when asset returns have unpredictable jumps

We study financial market incompleteness induced by discontinuities in asset returns. When there are multiple outcomes for a discontinuity, it is shown that this incompleteness cannot be removed by the introduction of extra securities. Claims cannot be hedged and are thereby not uniquely priced by arbitrage. We characterize the family of martingale measures associated with this form of incompleteness and discuss issues of existence and uniqueness for important special cases. Finally, using methods of stochastic control, we apply these results to derive replicating policies for arbitrary contingent claims and thereby relate the prices of contingent claims to the family of measures.