Title: Properties of the Wealth Process in a Market Microstructure Model
1Properties of the Wealth Process in a Market
Microstructure Model
- Ted Theodosopoulos
- Ming Yuen
- Department of Mathematics
- Drexel University
2INTERACTING 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
-
3RANDOMIZED NEIGHBORHOODS
- Each site is endowed with a random
neighborhood structure
-
which is a family of iid uniform random
variables taking values in
4STOCHASTIC 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
5AGGREGATE VARIABLES
Price process
Volume process
6AUXILLIARY VARIABLES
Volatility
7AUXILLIARY VARIABLES
are determined using the hypergeometric densities
8WEALTH 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
9WEALTH 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
10PATHS OF THE WEALTH PROCESS
11MARTINGALE 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.
12EXAMPLES OF TRANSITION RANGES
13STRATEGIC STABILITY
14ASYMMETRIC RISKS
15ASYMMETRIC RISKS
16ASYMMETRIC RISKS
excursions above
fluctuations in
excursions below
fluctuations in
17SOJOURN TIMES
18CONCLUSIONS
- 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.
19NEXT 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.