An infinite-dimensional model of liquidity in financial markets

Sergey V Lototsky; Henry Schellhorn; Ran Zhao

doi:10.3934/puqr.2021006

Article Contents

2021, Volume 6, Issue 2: 117-138. Doi: 10.3934/puqr.2021006

This issue Previous Article Dual representation of expectile-based expected shortfall and its properties Next Article Correlated squared returns

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

Author Bio: Henry.Schellhorn@cgu.edu; ran.zhao@cgu.edu
lototsky@usc.edu
lototsky@usc.edu

Received: September 2019

Accepted: May 17, 2021

Published: June 2021

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

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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.

Keywords:

Liquidity modeling,
Brownian sheet,
Itô-Wentzell formula,
No-arbitrage condition,
Stochastic partial differential equations.

Mathematics Subject Classification: Primary 91G80; 91B26; secondary 91G60.

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