Financial networks
Financial Networks
Introduction
Theme: discover a relationship among traded shares (equity)
look at historical market data to see whether price variations relate to each other.
Are there regularities that could anticipate the future behaviour of price?
In Food Networks (Ch. 1) we discovered a regularity:
\(\frac{\# pred}{\# prey} \approx 1\)
Important assumption
When markets are calm, investment becomes somewhat ``mathematical’’
Price time series
Proportional return on investment
depends on time
essentially, the discrete counterpart of the time derivative of price:
\[ r(\Delta t) = \frac{p(t_0 + \Delta t) - p(t_0)}{p(t_0)} \]
\[ r(\Delta t) = \frac{p(t_0 - \Delta t) - p(t_0)}{p(t_0)} \]
in the limit \(\Delta t \rightarrow 0\) it can be rewritten:
\[ r(t) \simeq \frac{d \ln{p(t)}}{dt} \]
For discrete time:
\[ r = \ln p(t_0 + \Delta t) - \ln p(t_0) \]
Correlation of prices
correlations in time series (or simply comovements) are valuable indicators
Two shares are correlated if historically their price varied in a similar way.
To qualify such a relation compute the correlation between their price returns over \(\Delta t\).
Let \(\langle r_i \rangle\) be the average return of i over \(\Delta t\)
\[ \rho_{ij}(\Delta t) = \frac{\langle r_i r_j\rangle - \langle r_i\rangle \langle r_j\rangle}{\sqrt{(\langle r_i^2\rangle - \langle r_i\rangle^2 )( \langle r_j^2\rangle - \langle r_j\rangle^2)}} \]
high \(\rho\)’s might uncover hidden links between stocks.
however, monitoring \(n(n-1)\) correlations quickly becomes unfeseable
we focus on high \(\rho\) values.
The Spanning tree of stocks
Similar-behaviour shares
Correlation (or lack of it) induces a distance b/w shares:
\[ d_{ij}(\Delta t) = \sqrt{2(1-\rho_{ij}(\Delta t))} \]
Let \(D(\Delta t)\) be the complete matrix of pairwise distances:
it describes a complete, weighted network!
Prune it to create its Mimimum Spanning Tree (MST)
The MST has only n-1 heavy connections while maintaining connectivity.
Resulting model
The MST of 141 NYSE high-cap stocks, \(\Delta t =\) 6h:30min
Some shares are hubs for local clusters of highly-correlated shares.
Consequences
Network analysis helps indentifying local clusters
Each clusters will have a “hub” share
Hub shares can signal the beaviour of the whole cluster:
they provide leads in forecasting how sections of the market will move.