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We consider a dynamic market model where buyers and sellers submit limitorders. If at a given moment in time, the buyer is unable to complete hisentire order due to the shortage of sell orders at the required limit price,the unmatched part of the order is recorded in the order book. Subsequentlythese buy unmatched orders may be matched with new incoming sell orders. Theresulting demand curve constitutes the sole input to our model. The clearingprice is then mechanically calculated using the market clearing condition. Weuse a Brownian sheet to model the demand curve, and provide some theoreticalassumptions under which such a model is justified.  Our main result is the proof that if there exists a unique equivalentmartingale measure for the clearing price, then under some mild assumptionsthere is no arbitrage. We use the Ito- Wentzell formula to obtain that result,and also to characterize the dynamics of the demand curve and of the clearingprice in the equivalent measure. We find that the volatility of the clearingprice is (up to a stochastic factor) inversely proportional to the sum of buyand sell order flow density (evaluated at the clearing price), which confirmsthe intuition that volatility is inversely proportional to volume. We alsodemonstrate that our approach is implementable. We use real order book data andsimulate option prices under a particularly simple parameterization of ourmodel.  The no-arbitrage conditions we obtain are applicable to a wide class ofmodels, in the same way that the Heath-Jarrow-Morton conditions apply to a wideclass of interest rate models.

A No-Arbitrage Model of Liquidity in Financial Markets involving Brownian Sheets