# American Institute of Mathematical Sciences

June  2021, 6(2): 117-138. doi: 10.3934/puqr.2021006

## An infinite-dimensional model of liquidity in financial markets

 1 Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA 2 Institute of Mathematical Sciences, Claremont Graduate University, Claremont, CA 91711, USA 3 Drucker School of Management, Claremont Graduate University, Claremont, CA 91711, USA

lototsky@usc.edu

Received  September 2019 Accepted  May 17, 2021 Published  June 2021

Fund Project: The authors gratefully acknowledge very helpful suggestions from the editors, referees, and the editorial staff.

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.

Citation: Sergey V Lototsky, Henry Schellhorn, Ran Zhao. An infinite-dimensional model of liquidity in financial markets. Probability, Uncertainty and Quantitative Risk, 2021, 6 (2) : 117-138. doi: 10.3934/puqr.2021006
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1There is no loss of generality in that statement. A buy market order can be specified in our model as a buy limit order with a limit price equal to infinity. Since we model assets with only positive prices, a sell market order can be specified in our model as a sell limit order with a limit price equal to zero.

2A notable exception is [10].

3We do not consider markets for swaps, where the price can be negative.

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