## Abstract

In this paper, we propose an artificial market to model high-frequency trading where fast traders use threshold rules strategically to issue orders based on a signal reflecting the level of stochastic liquidity prevailing on the market. A market maker is in charge of adjusting prices (on a fast scale) and of setting closing prices and transaction costs on a daily basis, controlling for the volatility of returns and market activity. We first show that a baseline version of the model with no frictions is able to generate returns endowed with several stylized facts. This achievement suggests that the two time scales used in the model are one (possibly novel) way to obtain realistic market outcomes and that high-frequency trading can amplify liquidity shocks. We then explore whether transaction costs can be used to control excess volatility and improve market quality. While properly implemented taxation schemes may help in reducing volatility, care is needed to avoid excessively curbing activity in the market and intensifying the occurrence of abnormal peaks in returns.

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## Notes

- 1.
This terminology, now quite common in the financial literature but possibly less used elsewhere, was actually introduced by the economist Nicholas Kaldor to refer to the most relevant elements requiring explanation. In his words, “The theorist...ought to start off with a summary of the facts which he regards as relevant to his problem [and] concentrate on broad tendencies, ignoring individual detail, and proceed on the ‘as if’ methods, i.e. construct a hypothesis that could account for these ‘stylized facts’...”, see Kaldor (1961).

- 2.
To facilitate the comprehension, we use capital letters to denote all intra-day variables, whereas small letters are used to denote daily variables. Accordingly, intra-day variables are indexed by

*t*, whereas calendar dates are indexed by*n*. - 3.
We assume that the demand is always fulfilled by a market maker, who adjusts taxation in order to limit/entice activity on the market.

- 4.
\({\mathbb {E}}^i[\cdot ]\) is the expectation with respect to the joint distribution of vector \(\underline{\epsilon ^{-i}}=(\epsilon _j)_{j\ne i}\).

- 5.
We stress that the market maker also operates on a fast scale: he adjusts prices on the fast intra-day time scale and, eventually, makes the closing daily price available for time series analysis.

- 6.
The truncation to positive values is just one of the possible positive transformations of the Ornstein–Uhlenbeck process. See, for instance, Almgren (2012) for other specifications of positive transformations for mean-reverting stochastic processes representing liquidity on markets.

- 7.
The Matlab code used to numerically simulate our returns, including the iteration procedure of the best response map, can be downloaded at https://drive.google.com/open?id=1Gx-UpwslWZPpPvabaARFtj5YSWpw4weg. The simulations produced by this code are exactly the ones that are statistically analyzed in Sect. 3.2.

- 8.
The level of \({\bar{\tau }}\) is not crucial, provided that it is large enough to reach a stable value for intra-day returns. A value of \({\bar{\tau }}\) that is too low would not let the intra-day market stabilize on an

*equilibrium*value for returns. - 9.
- 10.
We used the EuStockMarkets dataset, which is available in R Core Team (2017) and contains the time series of the DAX, the SMI, the CAC and the FTSE from the beginning of 1991 to the end of 1998 (1860 daily observations for each index) to compute summary statistics of the returns. For instance, standard deviations and kurtosis are 0.010, 0.009, 0.011, 0.008 and 9.28, 8.74, 5.39, 5.64 for the four time series, respectively. Similar figures are reported as examples for stocks and indices in Campbell et al. (1997), even though there is obviously considerable variability among different assets or financial acitvities. In Pagan (1996), the estimate of autocorrelation at lag 1 of squared returns for US stocks is 0.189.

- 11.
To run the numerical fit, we fix a seed for the generation of the random signals of the Ornstein–Uhlenbeck process. We then use a \(20\times 20\) grid of values for \(\mu \) and \(\sigma \). This calibration operation required about 30 h of machine time on a Core i7-6700 processor. The same methodology was implemented in a second round for the calibration of \(\theta \), where we used a grid of ten values ranging from 0.1 to 1. Note that implementation of a Monte Carlo experiment with \(M=20\) trajectories would require approximately 20 days. A statistical robustness check of this result follows from the analysis of time series produced by the model relying on those values of the parameters.

- 12.
Figures in Table 5 are computed as the median values extracted from the 19 simulations.

- 13.
We thank an anonymous referee for his or her useful remarks on tail exponents.

- 14.
As a proxy for volatility of daily returns, we have used the standard deviation of intra-day returns.

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## Acknowledgements

We are indebted to two anonymous referees for their helpful and constructive comments. The authors acknowledge the financial support of Ca’ Foscari University of Venice under the grant *“Interactions in complex economic systems: innovation, contagion and crises”*. We are also grateful to Antonella Basso for her support in sharing computer multi-core resources that proved to be invaluable for running our simulations. We received useful remarks from participants in the WEHIA 2017 conference held at Università Cattolica di Milano and seminars held at the University of Technology Sydney, Ca’ Foscari University of Venice and at the Centre d’Economie de la Sorbonne (CES), but are entirely responsible for all remaining errors.

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Liuzzi, D., Pellizzari, P. & Tolotti, M. Fast traders and slow price adjustments: an artificial market with strategic interaction and transaction costs.
*J Econ Interact Coord* **14, **643–662 (2019). https://doi.org/10.1007/s11403-018-0233-8

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### Keywords

- Artificial markets
- High-frequency trading
- Liquidity shocks
- Transaction costs