# PhD projects

## 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: Pricing outside the Black-Scholes paradigm

Usually we model agents as if they are maximisers of expected utility who act in a perfect market (perfect divisibility, zero transaction costs). I am interested in what happens if these assumptions break down.
###### 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.
###### Transaction costs
Consider an agent facing a portfolio choice problem who seeks to maximise expected utility from consumption but who faces transaction costs whenever he adjusts his porfolio. How should he act? Davis and Norman solved this problem for a single asset, but what happens as we move to multiple assets?
###### Behavioural finance
There is evidence agents do not maximise expected utility: instead they evaluate outcomes relative to reference points and they are risk seeking. Most interestingly from a mathematical perspective is the notion that they use non-linear expectations, often modelled by a transformation of probabilities which reweights events in favour of extreme outcomes. (It may be rational to buy lottery tickets, even though they have a negative expected return, if the perception of the rare event of a major win is overweighted.) When we move to dynamic models, it is a delicate issue to decide how best to model these phenomena, and to understand how agent behaviour changes.

### 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. Some of my latest work in this area involves American options

## Skills and Background

Most of my work involves modelling in continuous time, and the models are often based on Brownian motion or other diffusion processes. This is especially true of my work in pure stochastics, but also one of the common themes of my work in finance. Ideally, you should know about Brownian motion, martingales and Ito's formula, but these topics are not taught in all undergraduate programmes. If you wish to work on mathematical finance, then some background knowledge is useful, but most important is to have an interest in the topic. My aim is to use finance to generate new questions of probabilistic and mathematical interest.

## Next steps

The next step is to contact me by email: d.hobson "at" warwick.ac.uk It is useful to do this early, ie in the October-December before a start the following October, especially if you might need help with funding. Very limited funding is available for overseas students. The are more opportunities for UK and European students.