# An analysis of price impact function in order-driven markets

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3 148 G. Iori et al. / Physica A 324 (2003) Fig. 1. Schematic of the order-placement process. Stored limit orders are shown stacked along the price axis, with bids (buy limit orders) negative and oers (sell limit orders) positive. New limit orders are visualized as falling randomly onto the price axis. New oers can be placed at any price greater than the best bid, and new bids can be placed at any price less than the best oer. Market orders remove limit orders of the opposite sign, based on best price and earliest time. data, such as power-law placement frequency [7], and a power-law or log-normal order size distribution. 3. Results The depth prole n(p; t) gives the number of shares in the order book at price p and time t. We investigate how the depth prole depends on the parameters of the model (see Refs. [2,3] for further results predicted by the model regarding the behavior of prices and spreads). The simple technique of dimensional analysis allows us to derive several powerful results and simplify the simulation analysis by reducing the number of free parameters from ve to two. There are three fundamental dimensions in the model: shares, price, and time. There are also three rate constants:, with dimensions of shares=(price time),, with dimensions of shares=time, and, with dimensions of 1=time. There are two discreteness parameters: the order size and the price tick p 0. The average spread has dimensions of price and is proportional to =; this comes from a balance between the total order placement rate inside the spread and the order removal rate. The asymptotic depth is the density of shares far away from the midpoint, where market orders are unimportant. It has dimensions of shares=price, and is Poisson distributed with mean =. The slope of the depth prole near the midpoint has dimensions of shares=price 2. It is proportional to the ratio of the asymptotic depth to the spread, which implies that it scales as 2 =. This is summarized in Table 1. There is a unique way to combine the parameters to create nondimensional units for shares and prices as described in Table 2. When plotting the average depth proles, obtained by Monte

4 G. Iori et al. / Physica A 324 (2003) Table 1 Predictions of scaling of market properties as a function of properties of order ow. is the limit-order rate, is the market-order rate and is the spontaneous limit-order removal rate Quantity Dimensions Scaling relation Asymptotic depth shares=price d = Spread price s = Slope of depth prole shares=price 2 2 = = d=s Table 2 Nondimensional quantities and their denitions from observables with dimensions Description Notation Denition Depth ˆn n= Price ˆp p= granularity 2= n 0 nδ/α (a) p (b) p/p C Fig. 2. We plot the average depth prole for three dierent parameter sets both in (a) dimensional and (b) nondimensional units. The mean order size and the tick size p 0 set to one, and =0:5. p c is dened as =2. Three simulation results are shown, with =0:001 and =0:2, =0:002 and =0:4, =0:004 and =0:8. Carlo simulations for dierent values of and at x, in terms of nondimensional units, the curves collapse on each other (see Fig. 2). The functional form of the book depth-prole is primarily determined by the granularity parameter (Fig. 3a). In the high regime the market order removal rate is low and there is a signicant accumulation of orders at the ask (bid), so that the average depth n(p; t) is much greater than zero (note that price increments are calculated from the midprice). In this regime the depth is a concave function of prices. In the intermediate regime the market order rate increases, and n(p; t) decreases, almost linearly, toward zero at the

5 150 G. Iori et al. / Physica A 324 (2003) n/n C nδ/α and CDF (s/p C ) (a) p/p C (b) p/p C Fig. 3. (a)the mean depth prole, in nondimensional units, for three values of the granularity parameters, =0:2 (solid), =0:02 (dash) and =0:002 (dot). (b) A comparison of simulation results (thin solid line for depth prole and thin dashed line for cumulative distribution of the spread) with the self-consistent solution of the master equation approach (thick solid line for depth prole and thick dashed line for cumulative distribution of the spread) at = 0:02. ask. In the small regime the market order rate increases further, and the depth prole becomes a convex function close to the ask. We have studied the model analytically using a master equation approach and mean eld approximations to obtain the solution [2,3]. Analytical results in good qualitative agreement with the simulation both for the depth prole and the cumulative distribution of spreads can be obtained from one or another of these methods over a wide range of parameters (Fig. 3b). When the uctuations of the depth prole are small, a good approximation of the instantaneous price impact function is given by inverting the obvious relationship,! = p 0 n(p; t) dp : (1) As long as the depth prole is monotonically increasing,! is concave. Expanding n(p; t) in a Taylor series, one obtains terms p = (!; t; t)! from each order, with 6 1, which approximate the impact function over the ranges of price dominated at that order. This result is in agreement with the best available empirical evidence. The exponent 1 2 appears as a special case, when the depth prole satises the condition of vanishing at the midpoint and having a derivative that exists so that the depth increases linearly through a suciently wide range of prices. Even if our random ows model contains several unrealistic assumptions, it is because of its simplicity that we can relate its prediction to measurable properties of real markets. We nd that the price impact function is always concave, in agreement with the best available empirical measurements, and suggest that concavity emerges from the trading mechanism and market structure rather than rationally optimized trading strategies.

6 G. Iori et al. / Physica A 324 (2003) Acknowledgements We would like to thank the McKinsey Corporation, Credit Suisse First Boston, the McDonnell Foundation, Bob Maxeld, and Bill Miller for supporting this research. G.I. thanks the Santa Fe Institute for kind hospitality. References [1] F. Lillo, J. Doyne Farmer, Rosario N. Mantegna, Single curve collapse of the price impact function for the New York stock exchange, Nature 421 (2003) [2] Marcus G. Daniels, J. Doyne Farmer, Giulia Iori, Eric Smith, How storing supply and demand aects price diusion, [3] E. Smith, J. Doyne Farmer, L. Gillemot, S. Krishnamurthy, Statistical theory of the continuous double auction, [4] G. Iori, C. Chiarella, A simple microstructure model of double auction markets, Quantitative Finance 2 (5) (2002) [5] D. Challet, R. Stinchcombe, Analyzing and modelling 1+1d markets, Physica A 300 (2001) 285. [6] J.-P. Bouchaud, M. Mezard, M. Potters, Statistical properties of stock order books: empirical results and models, Quantitative Finance 2 (4) (2002) [7] Ilija I. Zovko, J. Doyne Farmer, The power of patience: a behavioral regularity in limit order placement, Quantitative Finance 2 (5) (2002)

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