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 and martingale optimal transport
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.
Recently there has been a lot of interest in the connections between the Skorokhod embedding problem and Martingale optimal transport. In optimal transport the goal is to transport mass from a given initial distribution to a target distribution in a way which minimises the average cost. In martingale optimal transport we add a conditon that mass must be transported in a way which respects the martingale condition. One example of a martingale transport is the left-curtain coupling introduced by Beiglboeck and Juillet. In joint work with Beiglboeck and Norgilas (a PhD student of mine) we show how to construct the left-curtain coupling for arbitrary probability measures.
Motivated by recent advances in martingale optimal transport, (and by the connection with mathematical finance, see below) the Skorokhod embedding problem continues to be an area of interest to probabilists. My approach is generally based on Brownian excursion theory which is a beautiful representation of Brownian paths.
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.
The Merton problem and extensions
In the Merton lifetime investment/consumption problem an agent seeks to maximise the expected discounted utility of consumption over the infinite horizon, where consumption is financed by investments in a risky asset. The problem is to determine the optimal consumption rate and the optimal level of investment in the risky asset. One approach to the problem is to derive the value function and to use the martingale optimality property, but to give a complete proof turns out to be surprisingly delicate. Having a full proof allows you to move on to extensions of the classical discounted utility framework, such as stochastic differential utility, much beloved by economists who believe it provides a better representation of agent preferences.
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.
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? Or, to preferences given by stochastic differential utility.
Liquidity and Poissonian optimal stopping
Typically we assume assets may be sold at a moment of the seller's choosing. But, what if the market is less liquid, and opportunities to sell are infrequent? One way to model this is to assume that opportunities to sell arise at event times of an independent Poisson process. What can we then say about the value function? The goal of my work is to try to understand how this model of illiquidity impacts on the prices of assets and the timing decisions of when to sell.
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.
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. There are more opportunities for UK students.