Power laws in market microstructure

Umut Çetin; Henri Waelbroeck

doi:10.3934/fmf.2023003

Article Contents

2023, Volume 2, Issue 1: 56-98. Doi: 10.3934/fmf.2023003 open access

This issue Previous Article Asset price bubbles, wealth preserving, dominating, and replicating trading strategies Next Article Data-driven nonparametric robust control under dependence uncertainty

Power laws in market microstructure

Umut Çetin^1, , and
Henri Waelbroeck^2,

1.
Department of Statistics, London School of Economics and Political Science, UK
2.
Baruch College, CUNY and AlgoCortex LLC, USA

^*Corresponding author: Umut Çetin
^*Corresponding author: Umut Çetin

Received: June 2022

Revised: September 2022

Early access: March 2023

Published: March 2023

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Abstract

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.

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

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Figure 1. The asymptotic behavior of $ F $ is shown for the case where signals are drawn from a truncated Gaussian distribution

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

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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

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

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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

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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

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

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

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

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

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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 $

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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

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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.

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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.

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