A theory for long-memory in supply and demand

Recent empirical studies have demonstrated long-memory in the signs of orders
to buy or sell in financial markets [2, 19]. We show how this can be caused by
delays in market clearing. Under the common practice of order splitting, large
orders are broken up into pieces and executed incrementally. If the size of
such large orders is power law distributed, this gives rise to power law
decaying autocorrelations in the signs of executed orders. More specifically,
we show that if the cumulative distribution of large orders of volume v is
proportional to v to the power -alpha and the size of executed orders is
constant, the autocorrelation of order signs as a function of the lag tau is
asymptotically proportional to tau to the power -(alpha - 1). This is a
long-memory process when alpha < 2. With a few caveats, this gives a good match
to the data. A version of the model also shows long-memory fluctuations in
order execution rates, which may be relevant for explaining the long-memory of
price diffusion rates.