Topology of Foreign Exchange Markets using
Hierarchical Structure Methods
Michael J. Naylor
1*
, Lawrence C. Rose
2
& Brendan J. Moyle
2
1
Department of Finance, Banking & Property
2
Department of Commerce
Massey University
Abstract
This paper uses two hierarchical techniques, a minimal spanning tree and an ultrametric
hierarchical tree, to extract a topological influence map for major currencies from the ultrametric
distance matrix. We find that these two techniques generate a defined and robust scale free
network with meaningful taxonomy, which is fundamentally different from that obtained from
stock market topology. The topology is shown to be robust with respect to method, to time horizon
and is stable during market crises. This topology gives a guide to determining the underlying
economic or regional causal relationships for individual currencies and will prove useful to
understanding the dynamics of exchange rate price determination as part of a complex network.
_______________________________________________________________________
PACS: 02.50.Sk, 89.65.s, 89.65.Gh, 89.75.Hc
Keywords: minimal spanning tree, ultrametric hierarchical tree, taxonomy, econophysics, financial markets
* Corresponding Author
Dept of Finance, Banking & Property
Massey University
Private Bag 11 222
Palmerston North
New Zealand
[email protected]
1
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1. Introduction
Hierarchical structure methods are used in finance is to ascertain the structure of asset price
influences within a market. These methods use the synchronous correlation coefficient matrix of
daily difference of log prices to quantify the pricing distance between assets in terms of the
inherent hierarchical structure. This structure will give some indication of the taxonomy of an
assets’ portfolio, and can be used to generate an asset markets' hierarchy.
Two techniques will be used in this paper. The first technique is the creation of a
minimal
spanning tree
(MST), which is a graph of a set of
n
elements of the arrangement of the nodes in an
ultrametric space
. MST has been shown to provide sound results for financial assets with the
resultant taxonomy displaying meaningful clusters [1, 2, 3, 4]. MST also helps to overcome the
empirical problem of noise in a historical correlation matrix [5].
The second technique is the creation of an
ultrametric hierarchical tree
structure [6, 7]. This
technique gives a determination of the hierarchical structure of a network and is particularly useful
for determining if hubs exist.
The structure of asset price movements is extracted by use of a synchronous correlation
coefficient matrix,
Aij
, of daily difference of log prices. This matrix is transformed [8] by the
equation below to get the ultrametric pricing distance between currencies. This is a superior metric
as it fulfils the three axioms of a metric distance [1].
)
1
(
2
)
,
(
ij
A
j
i
d
−
=
The choice of clustering procedure has more effect on the quality of clustering than the choice
of distance metric [9]. MST analysis uses the singlelinkage clustering method which builds up
clusters by starting with distinct objects and linking them based on similarity. The major issue
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 Winter '11
 BARNARD
 Physics, pH, Mass, Foreign exchange market, United States dollar, ISO 4217, USD, hierarchical tree

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