Mathematical Finance
Limit order books
The main focus of our research is creating models of limit order trading [1] in foreign exchange spot markets, where trades are conducted via bilateral trade agreements (rather than through a central counterparty). Under such a set-up, market participants are only able to trade with the subset of other market participants with whom they are bilateral trading partners.
The main difficulty with such modelling
is that each market participant views a local limit order book, whose
contents depend on the bilateral trade agreements of the individual
market participant, whereas the data details only the global limit order
book (which can be considered as the union of all local limit order
books). Therefore, the standard statistical analysis techniques
discussed in the literature (e.g., [2-3]) are unsuitable for this
application.
We have introduced a new method of measuring prices in such
markets, with a frame of reference that is relative to the most
recently traded prices in the market. This has enabled us to uncover
robust statistical regularities (both through time and between different
currency pairs), which we have then used as the basis of my models of
trading. Furthermore, we have identified several "stylized facts" [4]
of price formation in such markets. These stylized facts enable better
understanding of the process of price formation, and can also be used as
objective criteria against which to assess model output.
The key focus for future work will be on developing
models of limit order book state in such markets. Such models might
enable model-based inference about the latent structure of the network
of bilateral trade agreements, and would help clarify how much (or
little) of the global limit order book a typical market participant is
able to view. Furthermore, better understanding of the stylized facts
will help to explain how market participants make decisions when
assessing the market. We will be analysing other high-frequency data
from a range of different markets in order to formally test the
hypothesis that the distribution of extreme price movements follows a
power law with exponent approximately equal to 3 - a stylized fact often
referred to as "The Inverse Cubic Law" [5]. Using a recently proposed
modification to the Kolmogorov-Smirnov 2-sample test together with a
formal, maximum likelihood framework for estimation of the power-law
tail exponent, we aim to formally test this hypothesis at an exact
significance level.
For more information, please contact Martin Gould, Mason Porter, or Sam Howison.
Key references in this area
- M.D. Gould, M.A. Porter, S. Williams, M. McDonald, D.J. Fenn, S. Howison (2012). Limit order books, arXiv:1012.0349.
- J.P. Bouchaud, M. Mézard, and M. Potters (2012). Statistical properties of stock order books: empirical results and models, Quantitative Finance 2(4), 251-256.
- M. Potters and J.P. Bouchaud (2003). More statistical properties of order books and price impact, Physica A 324.1, 133-140.
- R. Cont (2001). Empirical properties of asset returns: stylized facts and statistical issues, Quantitative Finance 1, 223-236.
- P. Gopikrishnan, M. Meyer, L.A.N. Amaral, and H.E. Stanley (1998). Inverse cubic law for the distribution of stock price variations, The European Physical Journal B-Condensed Matter and Complex Systems 3(2), 139-140.
Financial Networks
The
turmoil witnessed in financial markets in recent years has
illustrated important links between seemingly disparate markets and a
high level of connectivity of the global financial system. These
interdependencies between financial institutions or assets are often
poorly understood and can have large and unforeseen consequences,
proving to be very important in providing insight into macro-economic
risk and large corporate risk.
Networks are used to represent complex systems of interacting entities. We are interested in investigating the structure and dynamics of financial networks (using data from HSBC). "Community detection" is an important tool in network analysis; it is used to cluster the data into densely connected groups and can reveal underlying structure in the network and detect functionalities or relationships between the nodes. In particular, we have been using new methods of network science developed specifically for community detection in time-dependent networks.
Some
challenges that arise in extracting communities from financial data
include choosing an appropriate network representation (choice of the
nodes and edges in the network), applying the method to the chosen
network model and interpreting the output of the method. Another
issue is also that of allowing overlap between the different
communities, for which no method has been developed for
time-dependent networks, and which is still unresolved at the level
of static networks.
To attempt to address some of these difficulties, we have been re-thinking some of the ideas in community detection for evolving networks, carrying out numerical experiments to attempt to extract robust community partitions and formulating null models to test the significance of the resulting partition.
Up to this point, we have been studying a dataset of financial assets from different markets using a signed, weighted, fully connected time-dependent correlation network. Although we have been able to extract communities that seem to be consistent with previous studies carried out on the same dataset and identify some important financial events across time, it seems that some of the features defined in the current community detection method need to be modified to account for signed edges.
The industrial partner for this project is HSBC.
For more information please contact Marya Bazzi or Mason Porter.
Key references in this area
- P. J. Mucha, T. Richardson, K. Macon, M. Porter, J-P. Onnela (2010). Community Structure in Time Dependent, Multiscale, and Multiplex Networks. Science 328(5980): 876-878.
- M. A. Porter, J-P. Onnela, P. J. Mucha. (2009). Communities in Networks. Notices of the American Mathematical Society 56(9) :1082-1097 & 1164-1166.
- D. J. Fenn, M. A. Porter, S. Williams, M. McDonald, N. F. Johnson, N. S. Jones (2011). Temporal Evolution of Financial Market Correlations. Physical Review E 84(2): 026109.
- D. J. Fenn, M. A. Porter, S. Williams, M. McDonald, N. F. Johnson, N. S. Jones (2009). Dynamic Communities in MultiChannel Data: An Application to the Foreign Exchange Market During the 2007-2008 Credit Crisis. Chaos 19(3): 033119.