Martingale Schr\"odinger Bridges and Optimal Semistatic Portfolios

TL;DR

This paper characterizes the minimal-entropy martingale measure in a two-period market with static options and dynamic stocks, linking it to utility maximization and providing explicit solutions under certain conditions.

Contribution

It introduces a duality between minimal-entropy martingale measures and exponential utility maximization, overcoming non-closedness issues of semistatic strategies.

Findings

Explicit solution for the optimal martingale measure Q*

Demonstrates existence of an optimal portfolio under technical conditions

Provides a dense subset of calibrated measures with desirable properties

Abstract

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 Schr\"odinger bridge Q*; that is, the minimal-entropy martingale measure among all models calibrated to option prices. This minimization is shown to be in duality with an exponential utility maximization over semistatic portfolios. Under a technical condition on the physical measure P, we show that an optimal portfolio exists and provides an explicit solution for Q*. This result overcomes the remarkable issue of non-closedness of semistatic strategies discovered by Acciaio, Larsson and Schachermayer. 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.