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# A dynamic model of the limit order book

• We consider an equilibrium model of the Limit Order Book in a stock market, where a large number of competing agents post "buy" or "sell" orders. For the "one-shot" game, it is shown that the two sides of the LOB are determined by the distribution of the random size of the incoming order, and by the maximum price accepted by external buyers (or the minimum price accepted by external sellers). We then consider an iterated game, where more agents come to the market, posting both market orders and limit orders. Equilibrium strategies are found by backward induction, in terms of a value function which depends on the current sizes of the two portions of the LOB. The existence of a unique Nash equilibrium is proved under a natural assumption, namely: the probability that the external order is so large that it wipes out the entire LOB should be sufficiently small.

Mathematics Subject Classification: Primary: 49K21, 49J21; Secondary: 91A13, 91A20, 91A60, 91B25.

 Citation: • • Figure 1.  Left: a distribution function for the random variable $X$, describing the size of the external order. Right: a possible shape of the limit order book. If the external order is a buy order with size $X>0$, all the stocks in the shaded region on the right (with area $= X$), will be sold. If the external order is a sell order for an amount $Y>0$ of stocks, all the buy orders in the shaded region on the left (with area $= Y$), will be executed

Figure 2.  A plot of the density function $\phi$, with data as in (39). In this case, solving (36)–(38) we find $p_A = 10.0831$, $p_B = 9.6097$, $\bar p = 9.8464$

Figure 3.  A plot of the functions $U(p)$ in (11) and (21), with data as in (39)

Figure 4.  The ask price $p_A$ is found by solving the Cauchy problem (107), (15), and finding the price at which $U = 0$. To estimate the rate at which $p_A$ changes with the boundary data $\overline p$, it is convenient to invert the role of the variables $U, p$, thus obtaining the linear ODE (109) for $p = p(U)$. The figure shows how $p_A$ changes when the value of $\overline p$ is increased

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