An infinite-dimensional model of liquidity in financial markets

Abstract

We consider a dynamic market model of liquidity where unmatched buy and sell limit orders are stored in order books. The resulting net demand surface constitutes the sole input to the model. We model demand using a two-parameter Brownian motion because (i) different points on the demand curve correspond to orders motivated by different information, and (ii) in general, the market price of risk equation of no-arbitrage theory has no solutions when the demand curve is driven by a finite number of factors, thus allowing for arbitrage. We prove that if the driving noise is infinite-dimensional, then there is no arbitrage in the model. Under the equivalent martingale measure, the clearing price is a martingale, and options can be priced under the no-arbitrage hypothesis. We consider several parameterizations of the model and show advantages of specifying the demand curve as a quantity that is a function of price, as opposed to price as a function of quantity. An online appendix presents a basic empirical analysis of the model: calibration using information from actual order books, computation of option prices using Monte Carlo simulations, and comparison with observed data.