Properties of the Wealth Process in a Market Microstructure Model

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Properties of the Wealth Process in a Market
Microstructure Model

• Ted Theodosopoulos
• Ming Yuen
• Department of Mathematics
• Drexel University

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INTERACTING PARTICLE SYSTEM

• State space X is the set of binary spin
  configurations on a lattice imbedded on a torus.

• Typical element of X is given by

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RANDOMIZED NEIGHBORHOODS

• Let
  

• Each site is endowed with a random
  neighborhood structure
  

which is a family of iid uniform random
variables taking values in

• Interaction potential

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STOCHASTIC PROCESS
is a continuous-time Markov process on X
for all with probability
are exponentially distributed epochs
The site x to flip is chosen uniformly
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AGGREGATE VARIABLES
Price process
Volume process
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AUXILLIARY VARIABLES
Volatility
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AUXILLIARY VARIABLES
are determined using the hypergeometric densities
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WEALTH PROCESS
We restrict our attention to the frozen phase of
the supercritical regime, and define the wealth
process for individual agents, and for the market
in aggregate
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WEALTH PROCESS
Using the invariant distribution of the
underlying Markov process we can construct the
following representations for the conditional
expectations of the wealth increments at the
aggregate and individual levels respectively
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PATHS OF THE WEALTH PROCESS
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MARTINGALE REPRESENTATION
There exist four integer functions
such that the aggregate wealth process is a
submartingale while the underlying Markov process
is in the region
These function satisfy
For large enough l, and the
lower interval disappears.
For small enough l, and the
two intervals merge.
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EXAMPLES OF TRANSITION RANGES
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STRATEGIC STABILITY
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ASYMMETRIC RISKS
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ASYMMETRIC RISKS
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ASYMMETRIC RISKS
excursions above
fluctuations in
excursions below
fluctuations in
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SOJOURN TIMES
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CONCLUSIONS

• There are instances of discontinuous jumps of the
  equilibrium configuration, as necessitated by
  qualitative changes to the submartingale ranges
  of the wealth process.
• Strategic stability analysis offers a behavioral
  interpretation of the invariant measure for the
  spin market model of market microstructure.
• The equilibrium configurations are generally
  asymmetric, and the asymmetry increases with
  price volatility. For low volatility,
  seller-surplus equilibria are feasible. As
  volatility increases, the system passes through a
  complex intermediate stage with two disconnected
  submartingale components, to arrive at a unique
  buyer-dominated equilibrium for sufficiently high
  volatility.

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NEXT STEPS

• Viability analysis of a strategic market
  microstructure game, Econophysics Colloquium
  (Canberra, November 2005)
• Stability analysis of set-valued equations which
  arise from allowing agents to flip their spins
  strategically.
• Periodic attractors of random truncator maps,
  SigmaPhi Conference (Crete, August 2005)
• Symbolic dynamics representation of the periodic
  orbits that make up the invariant measure of the
  process.