Optimal Trading in Financial Markets: a multi-pronged mathematical approach

The student will be investigating the role of Optimal Trading Algorithms in financial markets. They will need to identify new and emerging areas such as those in localised energy markets or crypto- currencies, where such models are applicable (and indeed required). Primarily we will be interested in the interactions between market participants and how individual strategies affect each other or the wider market. Furthermore, the results from the model could be used to form effective future regulation of said markets.

The student will need to develop and identify the major participants in the market and be able to describe the different roles they play. For each of the participants, the student will need to determine the key variables (and then parameters) affecting the market behaviour and build stochastic and/or deterministic models as appropriate. Further, the student will need to frame the behaviour of participants as the solution to optimal trading problems. Given the input variables and models, the problems will likely ultimately lead to Partial Differential Equations (PDEs), for which we expect the student must employ both analytical and numerical solution techniques. Since the decisions made to trade happen over very short timescales, whereas decisions to invest or regulate a market happen over much longer timescales, the student will need to be able to solve these problems over both short timescales and also examine their effect over the long term through a variety of scenarios, i.e. these are likely to be very much multi-scaled processes. Therefore some types of simulation techniques are likely to be required to explore the models, at least in the first instance, in order to guide the development of deterministic models.

In order that they will be able to build appropriate models and solve the resulting problems, the student will need to employ a variety of techniques. For example, simulation methods will first be tested on some classes of non-standard Stochastic Differential Equations to show how they can build up probabilistic scenarios of future events. We believe this should then guide the development of a novel finite-difference approach, that can be tailored to the equations to make them many times more efficient than standard simulation techniques. The student will be required to solve the resulting complex and novel PDEs, that could present new and interesting challenges. It is likely that existing techniques will need to be adapted or extended to provide accurate results for these complex problems in a timely manner, so that a full range of future scenarios can be generated