Econophysics Ancona 2007

1. Econophysics Ancona 2007 Network Centrality and Stock Market Volatility The Impact of Communication Topologies on Prices September 28th 2007 by Dr. Michael Schwind

2. Contents • Frankfurt Artificial Stock Market • Components • Retail Agents • Communication Network • Auction Method • Network Centralization • Degree Centralization • Betweeness Centralization • Closeness Centralization • Initial Networks • Results • Parameters • Simulation Run • Centralization and Volatility • Centralization and Distortion • Interpretation

3. FASM: Model Components • Bounded rational heterogeneous Agents: • Fundamental Agents (trade according exogenous inner value) • Trend Agents (trade according moving average) • Retail Agents (herd with threshold with direct neighbours) • Inter-Agent-Communication-Network: • Random-, Scale-Free-, Small World-, Star-Network • Auction Method: • Double-Auction with Limit-Order-Book

4. FASM: Retail Agents (1) • Retail Agents are initially not endowed with a trading strategy • They are able to adopt both trading strategies (trend, fundamental) • They are initially inactive and get activated by an individual price increase at the stock exchange • Once activated retail agents may adopt a trading strategy only from their direct neighbors within the communication network. Three cases are possible: • no neighbor with strategy no trading, wait • neighbor has strategy adopt and start trading • several neighbors with strategy adopt the best one and start trading

5. FASM: Retail Agents (2) • Retail agents stop trading and go into hibernation if an individual price decrease at the stock exchange occurred (e.g. 10%) • They sell all their shares over a defined period (e.g. 15 days) and remain inactive for an individual number of days (e.g. 200 days) • When the hibernation period is over, they may get activated again depending on their individual threshold

6. Retail Agent Behavior

7. Initial Distribution of Agent Types Green=Retail Agents, Yellow=Fundamental Agents, Red=Trend Agents

8. Distribution of Agent Types Green=Retail Agents, Yellow=Fundamental Agents, Red=Trend Agents

9. Full Activation of Retail Agents Green=Retail Agents, Yellow=Fundamental Agents, Red=Trend Agents

10. Double Auction Batch Limit Order Book Orders Possible Trade Volume The maximum possible trade volume defines the new price at 1019.

11. Degree Centralization The degree centralization measures the variation of the degree of a network member in relation to all other network members. (g=number of nodes, n*=node with highest degree) The degree centralization varies between 0 and 1. The star network has a degree-centralization of 1.

12. Betweenness Centralization Interactions between two nonadjacent nodes A and B depend on other nodes that exist on the path from node A to node B. The betweenness centralization measures the frequency of a node appearing on the path between the two nonadjacent nodes in relation to the other nodes of the network. pjk(i) equals the probability that node i is on the path between node j and k sjk equals the amount of shortest paths between nodes j and k. The betweenness centralization varies between 0 and 1, it reaches a maximum if a node is on all shortest paths between all other nodes (star network).

13. Closeness Centralization The closeness centralization measures how close a node is to the other nodes of a network in relation to the other nodes of the network. It shows how quickly (shortest paths to other nodes) one node can be reached from other nodes. d(i, j) being the distance (length of the shortest path) between node i and j.

14. Used Centralization Measures The high Betweeness Centralization of Small-World Networks may be due to the Small-World-Effect (?).

15. Initial Random Network Red=Retail Agents, Yellow=Fundamental Agents, Blue=Trend Agents

16. Initial Scale-Free Network Red=Retail Agents, Yellow=Fundamental Agents, Blue=Trend Agents

17. Initial Small-World Network Red=Retail Agents, Blue=Fundamental Agents, Yellow=Trend Agents

18. Initial Star Network Red=Retail Agents, Blue=Fundamental Agents, Yellow=Trend Agents

19. FASM: Model Parameters • Simulation Parameters: • Amount of trading days: 3,000 days (circa 12 years) • Inner Value: Random Walk • Communication Probability: 10% • Agent Parameters: • Number of Agents: 100 Agents (7 Fundamental, 7 Trend, 86 Retail Agents) • Assets: randomly distributed • Profit Horizon: 20 days • Communication-Network Parameters: • Topology: Random, Scale-Free, Star, Small-World • The parameters are held constant for the repetitions. Only the network types are varied.

20. Simulation Run

21. Number of Agent Types

22. Volatility and Distortion T=trading days(3,000), P=Price, Pf=fundamental value

23. Volatility and Network Centralization

24. Distortion and Network Centralization

25. Interpretation • Interpretation of Results: • The different network topologies affect the characteristics of the resulting time series of prices. • Increasing network centralization in terms of degree, betweeness and closeness seem to lead to increased volatility and distortion of prices. • Stock market models that allow herding of bounded rational agents should chose a proper communication network structure since it may be an important parameter.