  We use standard physics techniques to model trading and price formation in a
market under the assumption that order arrival and cancellations are Poisson
random processes. This model makes testable predictions for the most basic
properties of a market, such as the diffusion rate of prices, which is the
standard measure of financial risk, and the spread and price impact functions,
which are the main determinants of transaction cost. Guided by dimensional
analysis, simulation, and mean field theory, we find scaling relations in terms
of order flow rates. We show that even under completely random order flow the
need to store supply and demand to facilitate trading induces anomalous
diffusion and temporal structure in prices.
