Martingale Schrödinger bridges and optimal semistatic portfolios

In a two-period financial market where a stock is traded dynamically and European options at maturity are traded statically, we study the so-called martingale Schrödinger bridge Q ∗ $Q_{*}$ , that is, the minimal-entropy martingale measure among all models calibrated to option prices. This minimisation is shown to be in duality with an exponential utility maximisation over semistatic portfolios. Under a technical condition on the physical measure P $P$ , we show that an optimal portfolio exists and provides an explicit solution for Q ∗ $Q_{*}$ . This result overcomes the remarkable issue of non-closedness of semistatic strategies discovered by Acciaio et al. (Finance Stoch. 21:741–751, 2017). Specifically, we exhibit a dense subset of calibrated martingale measures with particular properties to show that the portfolio in question has a well-defined and integrable option position.