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

  1. M.D. Gould, M.A. Porter, S. Williams, M. McDonald, D.J. Fenn, S. Howison (2012). Limit order books, arXiv:1012.0349.
  2. 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.
  3. M. Potters and J.P. Bouchaud (2003). More statistical properties of order books and price impact,  Physica A 324.1, 133-140.
  4. R. Cont (2001). Empirical properties of asset returns: stylized facts and statistical issues, Quantitative Finance 1, 223-236.
  5. 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.