Dynamics of the FX Market: A Minimal Spanning Tree Approach

1. Dynamics of the FX Market:A Minimal Spanning Tree Approach Omer Suleman OCCF and Department of Physics University of Oxford Collaborators: N F Johnson, M McDonald, S Williams, S Howison

2. Networks Yeast Proteins World Wide Web Stock Market High School Dating

3. Networks Cyclic Network Tree: Acyclic Network Fully Connected Network

4. Networks of Financial Time Series • Correlation Based Networks • Entities generating financial time series (stocks, indices, hedge funds or currencies) are represented by nodes. • Weighted edges between nodes represent the correlation between the time series generated by these entities. • this gives us a fully connected network with ½[n(n-1)] edges where n is the number of nodes.

5. Filtering the Connections • The fully connected network contains too many connections, each with a range of possible weights, and hence too much information for it to be useful. • A filter has to be applied to this network in order to extract the most important links between the nodes thus clustering them. • Any scheme to do this will need a measure of distance or dissimilarity between nodes.

6. Distance • The weights of the links between nodes are based on the correlation between them. • The most intuitive measure of distance is the Euclidian distance between the time series: • This is a non-linear transformation of the correlation which gives a metric distance between nodes.

7. Metric Space: d(x,x) = 0 d(x,y) = d(y,x) d(x,z) ≤ d(x,y) + d(y,z) Ultrametric Space: u(x,x) = 0 u(x,y) = u(y,x) u(x,z) ≤ max{ u(x,y) , u(y,z) } Ultrametricity Ultrametric distance is a measure of distance found useful for data classification. • Many different Ultrametrics are possible on a space • Out of all Ultrametrics such that: u(x,y) ≤ d(x,y) the greatest is called the Subdominant Ultrametric which is unique and can be determined by a Minimal Spanning Tree.

8. Minimal Spanning Tree Tree: A connected graph without cycles is called a tree. Spanning Tree: A subgraph that is a tree and reaches out to all vertices of the original graph is called a spanning tree of the graph. Minimal Spanning Tree: Out of all possible spanning trees of a graph the one with minimum total edge weight is called the Minimal Spanning Tree of the graph.

9. MST in Finance – Equity Market Mantegna, J-P Onnela et. al.

10. MST in Finance – Hedge Funds Miceli and Susinno

11. MST and FX Market • Hedge fund profits and stock market returns can be measured in a single currency. • Nothing in the currency market is absolute. • Prices for a currency are quoted relative to another, usually USD. • How do we build the tree without missing out any currency?

12. Data Description • We look at XAU and 10 currencies USD, CAD, GBP, DEM, CHF, SEK, NOK, AUD, NZD and JPY from Jan 1993 to Dec 1994. • Thus we have hourly data points for 10 time series of the form USD/X. • We expand this set to all time series Xi/Xj possible in this group. • This gives us 110 different time series, with every currency represented in the network.

13. Trees of Hourly Data

14. Gold Cluster

15. AUD Cluster

16. Spurious Correlations? Triangle Effect Correlation of returns:

17. Comparison of Real and Random Trees Intersection of real and random MST for 1993-94 Currency MST for 1993-94

18. Degree Distributions

19. Dynamic MSTs

20. 1.00 0.90 0.82 100 200 300 400 500 0 Stability of MST Single step survival ratio dt

21. Multi-step Survival of Links 1.0 0 3000 4000 4500 1000 1500 2000 2500 500 0

22. Dynamics of JPY Cluster

23. Clustering Coefficient and Dynamics

24. Work in progress • We are currently applying this analysis to higher frequency data (5 min, tick data). We hope this will give us a real time picture of the market and indicate the currencies “in play”. • We are also investigating the effect of market news, both expected and unexpected, on the currency trees.

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