A No-arbitrage Model of Liquidity in Financial Markets involving Brownian Sheets: Applications to High-Frequency Data


Henry Schellhorn

Institute of Mathematical Sciences
Claremont Graduate University
Claremont, CA, 91711, USA

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Ran Zhao

Institute of Mathematical Sciences
Claremont Graduate University
Claremont, CA, 91711, USA

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We present a dynamic market model of liquidity where unmatched buy and sell limit orders are stored in order books. The resulting net demand curve constitutes the sole input to the liquidity model. We prove that generically there is no arbitrage in the model if the driving noise is a Brownian sheet. Under the equivalent martingale measure the clearing price is a martingale, and options can be priced under the no-arbitrage hypothesis. The no-arbitrage conditions we obtain are applicable to a wide class of models, in the same way that the Heath-Jarrow-Morton conditions apply to a wide class of interest rate models.

In our empirical analysis, we use limit order high-frequency data from the NYSE Arcabook. We calibrate a particular version of our liquidity model to this data. We then simulate the demand curve under this model and calculate option prices. In most cases, our model reproduces observed smiles. 


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