This thesis studies the problems of semistatic trading strategies in a discrete-time financial market, where stocks are traded dynamically and European options at maturity are traded statically. First, we show that pointwise limits of semistatic trading strategies are again semistatic strategies. The analysis is carried out in full generality for a two-period model, and under a probabilistic condition for multi-period, multi-stock models. Our result contrasts with a counterexample of Acciaio, Larsson and Schachermayer, and shows that their observation is due to a failure of integrability rather than instability of the semistatic form. Mathematically, our results relate to the decomposability of functions as studied in the context of Schrödinger bridges.
Second, we study the so-called martingale Schrödinger bridge 𝑄⁎ in a two-period financial market; 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⁎. 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.
| Identifer | oai:union.ndltd.org:columbia.edu/oai:academiccommons.columbia.edu:10.7916/dwj7-qg67 |
| Date | January 2023 |
| Creators | Zhao, Long |
| Source Sets | Columbia University |
| Language | English |
| Detected Language | English |
| Type | Theses |
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