Systemic risk in derivative markets: A graph-theory analysis

Systemic risk in derivative

markets: a graph-theory analysis

Objectives

•  Empirical study on systemic risk in derivative markets •  Integration as a necessary condition for systemic risk

to appear

•  Previous literature on integration

•  Influence of physical as well as derivative markets •  Three-dimensional approach

- Observation time - Spatial integration - Maturity

Selected markets

•  Choice directed by:

- Concerns about speculation in commodities and particularly in the crude oil market

- Development of bio fuels - Portfolio management :

- Commodities as a new class of assets

- Weight of the energy products in commodity indexes •  Three sectors :

- Energy products

- Agricultural products

1. Energy products

- Two crude oils (US and Europe) - Heating oil (US)

- Gas oil (Europe)

- Two natural gases (US and Europe) 2. Agricultural products

- Soy beans and soy oil (US) - Corn (US)

- Wheat (US)

3. Financial products - Gold

- Interest rates : Eurodollar - Exchange rates : EUR/USD

14 markets

> 655 000 daily futures prices

(settlement prices)

Statistical physics and graph-theory

•  Complex evolving system

•  Use of recent methods originated from statistical physics •  Graph-theory

•  Application to financial markets: Mantegna, 1999 •  Minimum Spanning Tree (MST) :

Filter the information given by the correlation matrix of price returns

Graph built on correlations transformed into distances The shortest path for a shock transmission

Methodology

•  Construction of MST

•  Topology of the filtered networks •  Evolution of the MST over time

Main results

1. Topology

- Star-like trees in the spatial and 3-D dimensions - Chain-like trees in the maturity dimension

Star-like organization dominates 2. Emerging taxonomy

- Trees organized around the three sectors of activity - Center of the graph : two crude oils

- Links between the agricultural and financial sectors

- Crude as best candidate for the transmission of shocks 3. Integration

- Increases in all dimensions

Section1. Minimum spanning trees :

a correlation based method

Section 2. Topology of the MST

Section 1. Minimum spanning trees

•  Synchronous correlation coefficients ρ of prices returns r :

with: F(t), futures prices at t - Correlation matrix C, (N x N)

From correlation to distances

•  Non linear transformation

•  Distances d defined as follows:

•  Distance matrix D, (N x N) •  Full connected graph

•  Represents all the possible connections between N nodes (vertices), with N-1 links (edges)

Minimum spanning tree

•  All the nodes of the graph are spanned •  No loops

•  Result: links of the MST are a subset of the initial graph •  The information space is reduced from (N(N-1)/2) to (N-1) •  In this study : shortest path linking all nodes

Section 2. Topology of the MST

•  Static analysis on the whole time period •  Spatial, maturity and 3-D

Spatial dimension

Finance Agriculture

Maturity dimension

Heating oil – Month 1 to 18

M1

M18

Three dimensions

Finance Agriculture

Allometric properties of the MST

•  Quantifying the degree of randomness in the tree •  The root is the node with the highest connectivity •  Starting from this root, two coefficients A_i and B_i

are assigned to each node i:

Section 3. Dynamical studies

3.1. Correlation coefficients 3.2. Node’s strength

3.3. Normalized tree’s length 3.4. Survival ratios

3.5. Pruning the trees

3.1. Mean correlations (a) and their variances (b)

(3-Dimensions)

3.2. Node’s strength

•  Full connected graph

•  The node’s strength S_i indicates the closeness of one node i to the others:

Energy

US EU

3.3. Normalized tree’s length

•  Sum of the lengths of the edge belonging to the MST:

•  The more the length shortens, the more integrated the system is

3.4. Survival ratios

•  Robustness of the topology over time

•  The survival ratio S_R refers to the fractions of edges in the MST, that survives between two consecutive

trading days:

3.5. Pruning the trees

•  Analysis of inter-market and inter-sectors reorganizations

•  Consider only the links between markets, whatever the maturity is considered

Energy

Finance

Conclusions

•  Topology of the graphs and its consequences for shock transmission

Star-like / Chain like •  Taxonomy of the graph

Crude oil at the center Connections intra / inter sector Maturities and connections (?) •  Robustness over time of the trees

Further studies

•  Empirical studies

- Enrichment of the graph :

Introduction of transaction volumes and open interest - Analysis in the maturity dimension

- Analysis of shocks