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
The conventions \((0,0]: = \emptyset \) and \([1,1): = \emptyset \) are used.
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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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DOI: https://doi.org/10.1007/s00780-022-00490-x