Advanced Search
Article Contents
Article Contents

Power laws in market microstructure

  • *Corresponding author: Umut Çetin

    *Corresponding author: Umut Çetin 
Abstract Full Text(HTML) Figure(12) / Table(2) Related Papers Cited by
  • We develop an equilibrium model for market impact of trades when investors with private signals execute via a trading desk. Fat tails in the signal distribution lead to a power law for price impact, while the impact is logarithmic for lighter tails. Moreover, the tail distribution of the equilibrium trade volume obeys a power law. The spread decreases with the degree of noise trading and increases with the number of insiders. In case of a monopolistic insider, the last slice traded against the limit order book is priced at the fundamental value of the asset reminiscent of [17]. However, competition among insiders leads to aggressive trading, hence vanishing profit in the limit. The model also predicts that the order book flattens as the amount of noise trading increases converging to a model with proportional transactions costs with non-vanishing spread.

    Mathematics Subject Classification: Primary: 91G45, 60J45; Secondary: 91B44.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  The asymptotic behavior of $ F $ is shown for the case where signals are drawn from a truncated Gaussian distribution

    Figure 2.  Equilibrium impact and shortfall for logit-normal signals for the case of an insider ($ N=1 $)

    Figure 3.  Convergence to the upper bound for $ M=1 $ when signals are drawn from a logit-normal distribution and shared with $ N=25 $ traders

    Figure 4.  Functional form of the equilibrium for Gaussian signals, for $ N=25 $

    Figure 5.  Equilibrium solutions for log-normal signals for the cases of an insider ($ N=1 $), and shared signals with $ N=2 $, $ N=25 $. The discontinuity of $ h(x) $ at the origin is the bid-ask spread

    Figure 6.  Functional form of the equilibrium for Log-normal signals, for $ N=25 $. The log-normal distribution is not symmetric and this results in a notable difference between the positive and negative branches for cost as a function of trade size

    Figure 7.  Aggregate investor profit per share for Log-normal signals

    Figure 8.  The spread is shown as a function of the number of informed investors, for log-normal signals with $ \sqrt{\Sigma}=10\% $

    Figure 9.  Equilibrium solutions for Student signals for the cases $ N=2 $, $ N=3 $ and $ N=25 $

    Figure 10.  Functional form of the equilibrium for Student signals for $ \alpha=3 $, $ N=25 $

    Figure 11.  Equilibrium solutions for same-price liquidation with log-normal signals for the cases of an insider ($ N=1 $), and shared signals with $ N=2 $, $ N=25 $

    Figure 12.  Equilibrium for Gaussian signals, comparing the same-price and trading desk liquidation models ($ N=1, 2, 25 $). Dotted lines represent the same-price liquidation equilibrium

    Table 1.  Distributions with power-law impact

    Distribution Density $ \rho^+ $
    Beta prime $ x^{\lambda-1}(1+x)^{-(\lambda +\alpha)} $ $ \left(\frac{N-1}{N}\alpha -1\right)^{-1} $
    Fréchet $ (x-\beta)^{-(1+\alpha)}\exp\{-\left(\frac{x-\beta}{s}\right)^{-\alpha}\} $ $ \left(\frac{N-1}{N}\alpha -1\right)^{-1} $
    Lomax $ \left(1+\frac{x}{\lambda}\right)^{-(\alpha+1)} $ $ \left(\frac{N-1}{N}\alpha -1\right)^{-1} $
    Pareto $ x^{-(\alpha+1)} $ $ \left(\frac{N-1}{N}\alpha -1\right)^{-1} $
    Student $ \left(1+\frac{x^2}{\alpha}\right)^{-(\alpha+1)/2} $ $ \left(\frac{N-1}{N}\alpha -1\right)^{-1} $
    In above probability densities are given up to a scaling factor and implicit constraints are enforced to ensure they are well defined with finite mean. Moreover, $N>\frac{\alpha}{\alpha-1}$ in all of the above.
     | Show Table
    DownLoad: CSV

    Table 2.  Distributions with logarithmic impact

    Distribution Density Asymptotics
    Exponential $ \exp(-\lambda x) $ $ \frac{N}{\lambda(N-1)}\log x $
    Gaussian $ \exp(-(x-\mu)^2/\Sigma) $ $ \sqrt{\frac{2\Sigma N}{N-1}}\sqrt{\log x} $
    Inverse Gaussian $ x^{-3/2}\exp\left(-\frac{\lambda (x-\mu)^2}{2\mu^2 x}\right) $ $ \frac{2N\mu^2}{\lambda(N-1)}\log x $
    Normal Inverse Gaussian $ \frac{K_1(\lambda \zeta(x))}{\pi \zeta(x)}\exp(\delta \gamma+\beta(x-\mu) $ $ \frac{N}{(N-1)(\lambda+\beta-1)}\log x $
    Weibull $ x^{d-1}\exp(-\lambda^p x^p) $ $ \left(\frac{N}{\lambda^p(N-1)}\right)^{1/p}(\log x)^{1/p} $
    In above probability densities are given up to a scaling factor and implicit constraints are enforced to ensure they are well defined with finite mean. Moreover, $\zeta(x):=\delta^2 +(x-\mu)^2$ for the Normal Inverse Gaussian distribution.
     | Show Table
    DownLoad: CSV
  • [1] R. AlmgrenC. ThumE. Hauptmann and H. Li, Direct estimation of equity market impact, Risk, 18 (2005), 58-62. 
    [2] K. Back, Insider trading in continuous time, The Review of Financial Studies, 5 (1992), 387-409.  doi: 10.1093/rfs/5.3.387.
    [3] N. Bershova and D. Rakhlin, The non-linear market impact of large trades: Evidence from buy-side order flow, Quantitative Finance, 13 (2013), 1759-1778.  doi: 10.1080/14697688.2013.861076.
    [4] B. BiaisP. Hillion and C. Spatt, An empirical analysis of the limit order book and the order flow in the paris bourse, the Journal of Finance, 50 (1995), 1655-1689.  doi: 10.1111/j.1540-6261.1995.tb05192.x.
    [5] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, vol. 27, Cambridge university press, 1989.
    [6] R. BloomfieldM. O'hara and G. Saar, How noise trading affects markets: An experimental analysis, The Review of Financial Studies, 22 (2009), 2275-2302. 
    [7] J.-P. Bouchaud, J. Bonart, J. Donier and M. Gould, Trades, Quotes and Prices: Financial Markets under the Microscope, Cambridge University Press, 2018.
    [8] S. Chakravarty, Stealth-trading: Which traders' trades move stock prices?, Journal of Financial Economics, 61 (2001), 289-307.  doi: 10.1016/S0304-405X(01)00063-0.
    [9] J. D. FarmerA. GerigF. Lillo and H. Waelbroeck, How efficiency shapes market impact, Quantitative Finance, 13 (2013), 1743-1758.  doi: 10.1080/14697688.2013.848464.
    [10] T. FoucaultO. Kadan and E. Kandel, Limit order book as a market for liquidity, The Review of Financial Studies, 18 (2005), 1171-1217. 
    [11] X. GabaixP. GopikrishnanV. Plerou and H. Eugene Stanley, Institutional investors and stock market volatility, The Quarterly Journal of Economics, 121 (2006), 461-504.  doi: 10.3386/w11722.
    [12] X. GabaixP. GopikrishnanV. Plerou and H. E. Stanley, A theory of power-law distributions in financial market fluctuations, Nature, 423 (2003), 267-270.  doi: 10.1038/nature01624.
    [13] L. R. Glosten, Is the electronic open limit order book inevitable?, The Journal of Finance, 49 (1994), 1127-1161.  doi: 10.1111/j.1540-6261.1994.tb02450.x.
    [14] R. L. GoettlerC. A. Parlour and U. Rajan, Equilibrium in a dynamic limit order market, The Journal of Finance, 60 (2005), 2149-2192. 
    [15] P. Gopikrishnan, V. Plerou, X. Gabaix and H. Eugene Stanley, Statistical properties of share volume traded in financial markets, Physical Review E, 62 (2000), p. R4493. doi: 10.1103/PhysRevE.62.R4493.
    [16] C. W. Holden and A. Subrahmanyam, Long-lived private information and imperfect competition, The Journal of Finance, 47 (1992), 247-270.  doi: 10.1111/j.1540-6261.1992.tb03985.x.
    [17] A. S. Kyle, Continuous auctions and insider trading, Econometrica, 53 (1985), 1315-1335.  doi: 10.2307/1913210.
    [18] F. LilloM. Szabolcs and J. D. Farmer, Theory for long memory in supply and demand, Physical Review E, 71 (2005), 066122. 
    [19] S. Nadtochiy, A simple microstructural explanation of concave pice impact, Available at SSRN, (2020), 3515008.
    [20] C. A. Parlour, Price dynamics in limit order markets, The Review of Financial Studies, 11 (1998), 789-816.  doi: 10.1093/rfs/11.4.789.
    [21] A. F. Perold, The implementation shortfall: Paper versus reality, Journal of Portfolio Management, 14 (1988), 4-9.  doi: 10.3905/jpm.1988.409150.
    [22] V. PlerouP. GopikrishnanL. A. Nunez AmaralM. Meyer and H. Eugene Stanley, Scaling of the distribution of price fluctuations of individual companies, Phys. Rev. E, 60 (1999), 6519.  doi: 10.1103/PhysRevE.60.6519.
    [23] M. Potters and J.-P. Bouchaud, More statistical properties of order books and price impact, Physica A: Statistical Mechanics and its Applications, 324 (2003), 133-140. 
    [24] K. Rock, The Specialist's Order Book and Price Anomalies, tech. rep., Harvard University, 1989.
    [25] I. Roşu, A dynamic model of the limit order book, The Review of Financial Studies, 22 (2009), 4601-4641. 
    [26] P. Sandås, Adverse selection and competitive market making: Empirical evidence from a limit order market, The Review of Financial Studies, 14 (2001), 705-734. 
    [27] A. Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics, 5 (1955), 285-309.  doi: 10.2140/pjm.1955.5.285.
    [28] N. Torre, BARRA Market Impact Model Handbook, Berkeley, 1997.
    [29] G. Vaglica, F. Lillo, E. Moro and R. N. Mantegna, Scaling laws of strategic behavior and size heterogeneity in agent dynamics, Physical Review E, 77 (2008), p. 036110. doi: 10.1103/PhysRevE.77.036110.
    [30] E. ZarinelliM. TreccaniJ. Doyne Farmer and F. Lillo, Beyond the square root: Evidence for logarithmic dependence of market impact on size and participation rate, Market Microstructure and Liquidity, 1 (2015), 15500045.  doi: 10.1142/S2382626615500045.
  • 加载中
Open Access Under a Creative Commons license




Article Metrics

HTML views(1586) PDF downloads(185) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint