recent developments in high frequency financial ... - Index of

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market order distribution that changes with the state of the book. One only has to
base the computation of the conditional upper tail expectation on a vector of state
variables F, i.e. calculate bE½mm j Qþk;
FŠ.
For the purpose of this study we focus
on the unconditional market order distribution and leave modeling the conditional
market order distribution as a topic for further research.
3.2.2 Average profit conditions
S. Frey, J. Grammig
To justify the marginal zero expected profit assumption, one implicitly assumes a
repetitive two phase trading process. In phase one, agents submit and cancel limit
orders until the book is free of (expected) profit opportunities and no agent wants to
submit, revise or cancel her order. Limit orders are sorted by price priority and,
within the same price tick, by time priority. When the book is such an equilibrium
the order book should display no ‘holes’, i.e. zero volumes in between two price
ticks. In phase two, a single market order of a given size arrives and is executed
against the equilibrium order book. After this event we go back to phase one,
during which the book is replenished again until equilibrium is reached and another
market order arrives and so forth. Can this be a reasonable description of a real
world trading process? The descriptive statistics on the trading and order
submission activity reported in Table 1 indicate a dynamic trading environment.
For a large stock, like Daimler Chrysler, we have on average over 3,000 trade
events per day, about 19,000 submissions of limit orders, of which over 80% are
canceled before execution. One could argue that such an active limit order trader
behavior indicates a thorough monitoring of the book which eliminates any profit
opportunities. This is quite in line with the theoretical framework. However, with
on average 10 seconds duration between trade events (for Daimler Chrysler) the
time to reach the new equilibrium after a market order hits the book and before a
new order arrives, seems a short span.
The marginal break even conditions can also be challenged by the following
reasoning. The conditions imply nonzero expected profits for limit order units that
do not occupy the last position of the respective price ticks. On the other hand, this
implies that the whole book offers positive expected profits for traders acting as
market makers. If market making provides nonzero expected profit opportunities,
then this would attract new entrants and the competition between these would-be
market makers ultimately eliminate any profit opportunities.
These considerations lead us to consider an alternative to the marginal profit
conditions which does not rely on the assumption that limit order traders
immediately cancel or adjust all their orders which show negative expected profit
on a marginal unit, and that also acknowledges the effect of market maker
competition on expected profits. For this purpose we retain most of the
assumptions of the Glosten/Såndas framework. However, instead of evaluating
the expected profit of the marginal profit for the last unit at each quote k, we assume
that the expected profit of the whole block of limit orders at any quote is zero. The
marginal zero profit condition is thus replaced by an ‘average zero profit
condition’. This assumption allows to differentiate between two types of costs
associated with the submission of a limit order, a fixed cost component, like order
submission and surveillance costs, and marginal costs (per share), like execution or
clearing fees and opportunity costs of market making. In the appendix we show that