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Martingale Schrödinger bridges and optimal semistatic portfolios

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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 Schrödinger bridge \(Q_{*}\), 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 \(P\), 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 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.

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Notes

  1. The conventions \((0,0]: = \emptyset \) and \([1,1): = \emptyset \) are used.

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Authors and Affiliations

  1. Departments of Statistics and Mathematics, Columbia University, 1255 Amsterdam Avenue, New York, NY, 10027, USA

    Marcel Nutz

  2. Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY, 10027, USA

    Johannes Wiesel & Long Zhao

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  1. Marcel Nutz
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  2. Johannes Wiesel
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  3. Long Zhao
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Correspondence to Johannes Wiesel.

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MN acknowledges support by an Alfred P. Sloan Fellowship and NSF Grants DMS-1812661, DMS-2106056.

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Nutz, M., Wiesel, J. & Zhao, L. Martingale Schrödinger bridges and optimal semistatic portfolios. Finance Stoch 27, 233–254 (2023). https://doi.org/10.1007/s00780-022-00490-x

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