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In a two-period financial market where a stock is traded dynamically andEuropean options at maturity are traded statically, we study the so-calledmartingale Schr\"odinger bridge Q*; that is, the minimal-entropy martingalemeasure among all models calibrated to option prices. This minimization isshown to be in duality with an exponential utility maximization over semistaticportfolios. Under a technical condition on the physical measure P, we show thatan optimal portfolio exists and provides an explicit solution for Q*. Thisresult overcomes the remarkable issue of non-closedness of semistaticstrategies discovered by Acciaio, Larsson and Schachermayer. Specifically, weexhibit a dense subset of calibrated martingale measures with particularproperties to show that the portfolio in question has a well-defined andintegrable option position.

Martingale Schr\"odinger Bridges and Optimal Semistatic Portfolios