We consider a dynamic market model where buyers and sellers submit limit orders . If at a given moment in time, the buyer is unable to complete his entire 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. Subsequently these buy unmatched orders may be matched with new incoming sell orders. The resulting demand curve constitutes the sole input to our model. The clearing price is then mechanically calculated using the market clearing condition. We model liquidity by considering the impact of a large trader on the market and on the clearing price. We assume a continuous model for the demand curve. We show that generically there exists an equivalent martingale measure for the clearing price , for all possible strategies of the large trader, if the driving noise is a Brownian sheet, while there may not be if the driving noise is multidimensional Brownian motion. Another contribution of this paper is to prove that, if there exists such an equivalent martingalemeasure, then, under mild conditions, there is no arbitrage. We use the Ito-Wentzell formula to obtain both results. We also characterize the dynamics of the  demand curve and of the clearing price in the equivalent martingale measure. We find that the volatility of the clearing price is inversely proportional to the sum of buy and sell order flow density (evaluated at the clearing price ), which confirms the intuition that volatility is inversely proportional to volume. We also demonstrate that our approach is implementable. We use real order book data and simulate option prices under a particularly simple parameterization of our model. 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 .
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 prove that generically there is no arbitrage in the model when the driving noise is a stochastic string. Under the equivalent martingale measure , the clearing price is a martingale, and options can be priced under the no-arbitrage hypothesis. We consider several parameterized versions of the model, and show some advantages of specifying the demand curve as quantity as a function of price (as opposed to price as a function of quantity). We calibrate our model to real order book data , compute option prices by Monte Carlo simulation, and compare the results to observed data .
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where buyers and sellers submit limit orders . If at a given moment in time, the buyer is unable to complete his entire 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. Subsequently these buy unmatched orders may be matched with new incoming sell orders. The resulting demand curve
of liquidity where unmatched buy and sell limit orders are stored in order books. The resulting net demand surface
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our model. The clearing price is then mechanically calculated using the market clearing condition. We model liquidity by considering the impact of a large trader on the market and on the clearing price. We assume a continuous model for the demand curve. We show that generically there exists an
the model. We prove that generically there is no arbitrage in the model when the driving noise is a stochastic string. Under the
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for
,
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, for all possible strategies of the large trader, if the driving noise is a Brownian sheet, while there may not be if the driving noise is multidimensional Brownian motion. Another contribution of this paper is to prove that, if there exists such an equivalent martingalemeasure, then, under mild conditions, there is no arbitrage. We use the Ito-Wentzell formula to obtain both results. We also characterize the dynamics of
is a martingale, and options can be priced under
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no-arbitrage hypothesis. We consider several parameterized versions of the model, and show some advantages of specifying the
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and of the clearing price in the equivalent martingale measure. We find that the volatility of the clearing price is inversely proportional to the sum of buy and sell order flow density (evaluated at the clearing price ), which confirms the intuition that volatility is inversely proportional to volume. We also demonstrate that our approach is implementable. We use
as quantity as a function of price (as opposed to price as a function of quantity). We calibrate our model to
R
and simulate option prices under a particularly simple parameterization of our model. 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
, compute option prices by Monte Carlo simulation, and compare the results to observed data