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The following projects are all of theoretical character. 1. Convergence of multi-period binomial models towards continuous-time models: detailed study of one of the sections in Rachev and Ruschendorf, Theory Probab. Appl. 39 (1994), no. 1,120-152 (1995). 2. Convergence of option prices and replicating strategies in discrete-time models as the number of steps tends to infinity and (physical) time is fixed (e.g. Duffie-Protter 1992). 3. American option pricing by martingale methods: detailed study of the connection between optimal stopping in continuous time, Snell envelopes, and American option pricing (e.g. developing the arguments in Karatzas-Shreve). 4. Dynamic programming for portfolio optimization: in the Markovian setting one can associate to a portfolio optimization problem a partial differential equation that the value function is expected to solve, in a suitable sense. One may investigate existence, uniqueness and regularity of solutions to such PDEs. Some familiarity with PDEs is desirable. Starting point could be Fleming-Soner. 5. Proof of the fundamental theorem of asset pricing, in the setting of a general probability space and discrete time, following Kabanov and Kramkov, Theory Probab. Appl. 39 (1994), no. 3, 523-527 (1995). 6. Quantile hedging in discrete time, following the book by Rillmer and Schied.
Sovereign debt portfolios, bond risks, and the credibility of monetary policy: study of the article with the same title in J. of Finance, 75(6), 2020. Issues: correctness of statistical procedures, sensitivity of results to assumptions. Nota bene. Students are welcome to propose topics of their choice, or at least to ask for a topic in a certain area. I cannot guarantee though that such arrangements can always be made. Helpful advice on writing mathematical texts, that I strongly recommend to read carefully, can be found here.
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