Jochen Theis Dissertation Topic Ideas

If you find any of these topics interesting Jochen Theis is happy to discuss them in more detail with you. Please email mathfin [-at-] maths [dot] ox [dot] ac [dot] uk and I will put you in touch with Jochen.

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In interest rates the most interesting topics (apart from models with jumps and incomplete markets, which still have fundamental questions about the ability to hedge around them) are either linked to stochastic volatility or to particular numerical methods. There are basically two types of approaches generally followed for stochastic volatility: either one takes a market model and uses a stochastic volatility process for the individual forward rates (LIBORs), or one starts from a low-factor Markov-type model (a short rate model or a Markov functional model) and attempts to make the model's volatility process stochastic without changing the low-factor Markovian nature of the process. A third approach would be to add stochastic volatility to a general HJM-type model; this is the most general approach but also the least tractable in practice.

In the market model approach, one ends up with a very high-dimensional model that needs to be simulated with Monte Carlo methods, and the main problems are to find efficient Monte Carlo algorithms (not that complicated to come up with something, but tricky to do well), to find fast and efficient pricing methods for calibration instruments (harder) and to find stable and reliable calibration procedures (very hard).

There are basically two main popular approaches; one is to make the stochastic volatility simple to handle, i.e. to make the stochastic volatility process uncorrelated to the processes driving the forward rates and to control skew by adding a local volatility function (usually either CEV or displaced diffusion). The most popular of these is the model described by Piterbarg in several papers. The other approach is to use the same type of process that people use to generate volatility surfaces, namely SABR, as driving processes for the forward rates; this is results in the SABR market model, which has been promoted by Riccardo Rebonato and collaborators recently. I don't know of any successful use of this in practice, as it is very hard to calibrate reliably but it could be an interesting thesis topic to look at some aspects of the model.

The low-factor approaches are primarily interesting because they have very efficient numerical implementations. The most interesting topic here is to approximate market models with stochastic volatility by low-factor Markov models as a more efficient way to implement such models for specific cases. This is particularly relevant for calibration; the technique is sometimes called "Markovian projection". It could be interesting to look at this for the SABR market model.

There are simpler but very interesting questions related to this, which could also make for nice thesis topics. One is the pricing of spread options under the various models. In the markets spread options are often still priced with Black-type models, because one does not see much of a difference when shifting to stochastic volatility models with a single driving factor for volatility. However, the pricing does change when one has two factors driving stochastic volatility, e.g. a Heston or SABR type process for each of the two rates in the spread option. I think it would be a nice thesis topic to investigate spread option pricing under various different types of options. There is a slight risk here, though, that there may be something published on this that I don't know about.

A more practical set of problems is around calibration of market models. The basic issue is that the models have a very rich structure of parameters which can be calibrated in various different ways. The most popular is the so-called cascade algorithm, another is the use of a parametric time-homogeneous form for the forward volatilities (Rebonato's favourite). Investigation one of these for a stochastic volatility model or comparing the techniques could be a good thesis topic.

Topics around numerical methods are either around constructing efficient and stable Monte Carlo simulations, or around constructing efficient algorithms for low-dimensional models (PDEs, grids). For Monte Carlo, one interesting topic could be algorithms to simulate SABR (there has been a lot of work on the Heston process already, so there's little left to do there), and by extension the SABR market model. Another is the calculation of upper bounds for Bermudan callable products; there are well understood and quite efficient methods that give lower bounds, but it's hard to get good upper bounds (I discussed some of this in module 5).

For the low-dimensional models the most interesting implementation methodology is based on explicit semi-analytical integration of piecewise polynomial functionals of Brownian motion (also known as the SALI tree method). I supervised a thesis on the stability aspects of this recently (by Michael Toepler, submitted last year), which may give you some idea of what the method involves. There are some questions around this that I think would make for an interesting thesis topic.

Summing up, if you want to write on interest rate models, one could

* Discuss the SABR market model and do some work on its calibration

* Do a comparative study of the Piterbarg and SABR market models

* Investigate "Markovian projection" for some stochastic volatility model

* Study some calibration methods for market models, including stochastic volatility

* Investigate efficient Monte Carlo simulation methods of SABR and SABR market models

* Examine upper bound methodologies for Bermudan callable product under Monte Carlo

* Investigate some numerical aspects of typical implementation methods for Markov Functional models

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