Mathematical Finance: Complete markets and risk-neutral pricing Theoretical

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Subject：Mathematical Finance

Project/Topic： Complete markets and risk-neutral pricing Theoretical

 The Second Fundamental Theorem of Asset Pricing states that an arbitrage- free market model is complete if and only if there exists exactly one equiva- lent martingale measure. One goal of the project is to review this result for market models in discrete time. Further, pricing in incomplete markets, geometric characterisations of equivalent martingale measures and the re- lation to the martingale representation theorem will be considered. In a second part some of those concept will also be discussed in a continuous time setting.

Report/words count: 50 pages

Aim of project: Take hahn-banach as the key research object.

concerning the outline, in general the thesis should start with an introduction motivating the problem, a (formal or informal) description of the topic and the results in your own words. This is usually followed by a section 'Preliminaries', where the mathematical objects are carefully introduced and essential required results are recalled (in your case, for instance, martingales or the Hahn-Banach theorem among other things), and then the more technical sections begin (one could also put this in an appendix). Don't forget to mention other people's work and to include a list of reference.

Anyway, reorganising and restructuring the material is part of the task, so I leave it to you to come up with a first suggestion. But I am happy to discuss the basic outline with you. What about Wednesday at 10 am (UK time)? If that's fine with you, I will send you a zoom link.

By the way, the proof of one of the important theorems for your thesis relies on an application of the Hahn-Banach separation theorem, which is recalled in the appendix of Foellmer/Schied. Do you need a copy of the appendix or do you know about this theorem? (In case you have taken some classes in functional analysis you certainly know it)
Also Lemma 1.60 of the book is needed, stating that a certain subspace of L^1 is closed, but it would be fine for you just to quote it. Anyway, let me know if you need a copy of Chapter 1.

Tool: Best using matlab or visual. Pyhson is ok

What else: Must use latex for the whole paper

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