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10.2 The Standard and Widely Adopted Methodology
The methodology which is widely adopted in the literature stems from Mantegna’s
seminal paper [89] (cited more than 1550 times as of 2019) and Chap. 13 of the
book [90] (cited more than 4400 times as of 2019) published in 1999. We describe
it below:
• Let N be the number of assets.
• Let P i (t) be the price at time t of asset i, 1 ≤ i ≤ N .
• Let r i (t) be the log-return at time t of asset i:
r i (t) = log P i (t) − log P i (t − 1).
• For each pair i, j of assets, compute their correlation:
ρ i j =
r i r j − −r i r j
r
2
i − −r i 2
r
2
j − −r j 2
.
• Convert the correlation coefficients ρ i j into distances:
d i j =
2(1 − ρ i j ).
• From all the distances d i j , compute a minimum spanning tree (MST) using, for
example, Algorithm 1:
Algorithm 1 Kruskal’s algorithm
1: procedure BuildMST({d i j } 1≤i, j≤N )
2:
Start with a fully disconnected graph G = (V, E)
3:
E ← ∅
4:
V ← {i} 1≤i≤N
5:
Try to add edges by increasing distances
6:
for (i, j) ∈ V 2 ordered by increasing d i j do
7:
Verify that i and j are not already connected by a path
8:
if not connected(i, j) then
9:
Add the edge (i, j) to connect i and j
10:
E ← E ∪ {(i, j)}
11:
G is the resulting MST
12:
return G = (V, E)
Several other algorithms are available to build the MST [62].
The methodology described above builds a tree, i.e. a connected graph with N − 1
edges and no loop. This tree is unique as soon as all distances d i j are different. The