Potential projects for PhD students

I am always interested to hear from well qualified prospective PhD students interested in working in Probability Theory, Mathematical Finance, or at the interface between these two areas. Below are some examples of the types of projects which might be interesting to study.

Skorokhod embeddings

Let B be a Brownian motion and let p be a probability measure. The Skorokhod embedding problem is to find a stopping time T such that B(T) has law p. There are many solutions to the Skorokhod embedding problem, often based on auxiliary processes such as the running maximum or the local time at zero. There are also many interesting questions in this area - how to extend these stopping rules to other processes, and how to design stopping rules with specified optimality properties being just two examples.

Mathematical Finance: Optimal timing for an asset sale

Consider the following problem: you have an asset for sale, and you are free to choose the time at which to sell this asset. The asset is indivisible (this is not a common assumption in derivative pricing, but is appropriate in many real examples elsewhere in finance, such as company takeovers) and the asset sale is irreversible. No dynamic trading is possible in this asset, however you are free to invest on a financial market which includes assets which are partially correlated with the asset for sale. The issues are; how to formulate this problem (for example as an optimal consumption problem) and how to solve it to give an investment/consumption strategy and an optimal sale time.

Mathematical Finance: Robust bounds for derivative prices

The standard approach in mathematical finance is to postulate a model (and perhaps to calibrate this model using options data) and to use this model for pricing and hedging. The quality of the prices and hedges depends crucially on the quality of the model. Another approach is to consider the class of all models and then to reduce this class by considering only those models which exactly match the price of (liquidly) traded derivatives. A range of prices for (nontraded) exotic options can now be found by searching over those models which remain feasible. The advantage of this approach is that it gives robust, model-independent bounds on option prices. The disadvantage is that these bounds may be quite wide. They are however the tightest bounds which can be derived without introducing any modelling assumptions. Producing bounds across models often requires some sort of coupling of stochastic processes.