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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.

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The problem called debt overhang has recently been observed in international financial relations between a sovereign country and foreign commercial banks. The term “debt overhang” expresses the situation where a sovereign country has borrowed money from foreign banks and has been unable to fulfill the scheduled repayments for some time. We formulate this problem as a noncooperative game with the lender banks as players where each decides either to sell its loan exposure to the debtor country at the present price of debt on the secondary market, or to wait and keep its exposure. We propose two approaches: a one-period approach (Chapter II), and a direct dynamic approach (Chapter III). In the one-period approach, we consider a representative period, while in the dynamic approach, the whole dynamics is directly considered. Both approaches are consistent and complementary in that the first approach considers the effect of a large number of banks, and the second approach captures the dynamic nature of the problem. In the one-period approach, we consider the behavior of many banks. In the model with n lender banks, there are many pure and mixed strategy Nash equilibria. However we show that in any equilibrium, the resulting secondary market price remains almost the same as the present price when the number of banks is large. In addition, we discuss the structure of the set of Nash equilibria. The second approach is a direct dynamic formalization of the same problem with two creditor banks. We show that in the dynamic game there exist three types of subgame perfect equilibria with the property called the time continuation. We consider the relationships between the equilibria of the dynamic game and those of the one-period approach and show that the one-period approach does not lose much of the dynamic nature of the problem. In every equilibrium, each bank waits in every period with high probability, and this probability is close to 1 when the interest rate is small. If the price function of debt is approximated by some homogeneous function for large values of debt, then the central equilibrium probability becomes almost stationary in the long run. The stationary probability is relatively high as long as the interest rate is low. Finally, in Chapter IV, we consider the duration of debt overhang with two lender banks. We show that the equilibrium duration of debt overhang converges to a constant when the length of a subperiod tends to zero. The constant is large when the degree of homogeneity of the price function is high. When the degree of the homogeneity is low, the constant is close to In 2/ In β², where β is the annual interest factor. These results as a whole are interpreted as a tendency for the problem of debt overhang to persist over a long time. / Ph. D.