# How Complexity and Uncertainty Grew with Algorithmic Trading

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Rise of Algorithmic Trading

## 3. Hypothesized Effects

#### 3.1. Complexity (H1)

**Hypothesis**

**1**

**(H1).**

#### 3.2. Uncertainty (H2)

**Hypothesis**

**2**

**(H2).**

## 4. Data

#### 4.1. Dependent Variables

_{200}]) and then merge ten consecutive bins on a higher level, to get another more coarse-grained binning with 20 bins (which we denote with the subscript [_

_{20}]). One can think of the more fine-grained setup as trading on one more digit behind the dollar (e.g., distinguishing between a hundredth or a thousand of a cent).

#### 4.1.1. Complexity

#### 4.1.2. Remaining Uncertainty

#### 4.1.3. Derivation of Measures

_{eM}] (using the spectral decomposition [65]). We confirm the results for E and h with estimates derived with the Python package dit [66], which uses another derivation method, more directly frequency-based, and denote them with subscript [_

_{fq}] [48]. We also used cross-validation and other comparisons to test out different sampling-window length, being cautious with overfitting. Eventually, simplifying computational demands, we used window length of L = 2 for the method [_

_{eM}], and L = 4 for the less computationally demanding derivation method [_

_{fq}].

#### 4.2. Independent Variables

#### 4.2.1. Algorithmic Trading

_{lin}) and an exponential trend line (AT

_{exp}), and extrapolate the empirical average with a linear tendency for 2016–2017 (which we denote with AT

_{emp}). This gives us three general estimates for the increase of the level of algorithmic trading in FX markets

_{emp}), but we also test for the linear and exponential tendencies. It turns out that our results are very robust regardless of the choice among those three options. In other words, the resulting effects are very similar with any tendency that has monotonically increased over the decade, growing from less than 15% before 2007 up to some 70% in 2018. Since we are not aware of any other relevant variable that has shown such substantial and monotonic growth during this period, we increasingly became less worried about the precise shape of the empirical rise of algorithmic trading. This makes us quite confident to suggest that the role of any such monotonically increasing tendency represents some aspect of the growing involvement of algorithmic trading. At the least, it allows us to draw comparative conclusions in the context of our control variables (see below), none of which shows such monotonically increasing tendency. In other words, we detect effects of something other than the usual economic indicators that has substantially and monotonically increased over the decade. For lack of any other reasonable candidate for this variable, we argue that this effect stems from the increasing involvement of algorithmic trading.

#### 4.2.2. Control Variables

#### 4.2.3. Variable Versions

- Three dependent variables (DVs). We derive different versions of our three summary measures, namely from ϵ-machines (epsilon-machine,
_{eM}) and from frequency counts (_{fq}), calculated on basis of coarse-grained 20 bins (_{_20}) and more fine-grained 200 bins (_{_200}).- ○
- predictable information (E
_{eM_20}, E_{fq_20}and E_{eM_200}, E_{fq_200}) - ○
- predictive complexity (C
_{eM_20}and C_{eM_200}) - ○
- remaining uncertainty (h
_{eM_20}, h_{fq_20}and h_{eM_200}, h_{fq_200})

- Six independent variables (IVs). Our main variable of algorithmic trading is estimated in three different ways (see Figure 3).
- ○
- algorithmic trading
- ▪
- empirical with linear extrapolation (AT
_{emp}) - ▪
- linear tendency (AT
_{lin}) - ▪
- exponential tendency (AT
_{exp})

- ○
- lagged dependent variable (dep
_{t−1}) - ○
- GDP growth rate (GDPr)
- ○
- inflation rate (infl)
- ○
- interest rate (intr)
- ○
- unemployment rate (unpl)

## 5. Results

#### 5.1. Increasing Complexity (H1)

_{_20}) and more fine-grained 200 bins (

_{_200}). For reasons of didactic presentations, Figure 4 showcases the results only for the [_

_{eM}] derivation method and the empirical AT

_{emp}estimate. We provide the full tables for all 18 models in Supporting Information SI.3. The version visualized in Figure 4 is among the models where AT has the least influence, and is therefore a rather conservative estimate of our broader results.

#### 5.2. Decreasing Uncertainty (H2)

_{eM}] derivation method and the empirical AT

_{emp}estimate (for the full array of all 12 models, see Supporting Information SI.4). All models support the conclusions presented here.

#### 5.3. Robustness of Results for Complexity and Uncertainty (H1 & H2)

_{eM}] and [_

_{fq}] derivation methods. The confidence intervals (see error bars at estimated marginal means) clearly do not overlap, which shows that the dynamics have changed significantly in the indicated directions.

## 6. Discussion and Interpretation

#### 6.1. There’s Plenty of Room at the Bottom

#### 6.2. Digging Deeper: The Chain Rule of Entropy

#### 6.3. The More You Know, the More Uncertain You Get

## 7. Conclusions

#### 7.1. Infinitely More Levels of Uncertainty?

#### 7.2. Limitations and Future Outlooks

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Agrawal, A.; Gans, J.; Goldfarb, A. Prediction Machines: The Simple Economics of Artificial Intelligence; Harvard Business Press: Brighton, MA, USA, 2018. [Google Scholar]
- Brookshear, J.G. Computer Science: An Overview, 10th ed.; Addison Wesley: Boston, MA, USA, 2009. [Google Scholar]
- Johnson, N.; Zhao, G.; Hunsader, E.; Qi, H.; Johnson, N.; Meng, J.; Tivnan, B. Abrupt rise of new machine ecology beyond human response time. Sci. Rep.
**2013**, 3, 2627. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Farmer, J.D.; Skouras, S. An ecological perspective on the future of computer trading. Quant. Financ.
**2013**, 13, 325–346. [Google Scholar] [CrossRef] - Golub, A.; Dupuis, A.; Olson, R.B. High-Frequency Trading in FX Markets. In High-frequency Trading: New Realities for Traders, Markets and Regulators; Easley, D., Prado, M.M.L.D., O’Hara, M., Eds.; Risk Books: London, UK, 2013; pp. 21–44. ISBN 978-1-78272-009-6. [Google Scholar]
- Mahmoodzadeh, S. Essays on Market Microstructure and Foreign Exchange Market. Ph.D. Thesis, Simon Fraser University, Department of Economics, Burnaby, BC, Canada, 2015. [Google Scholar]
- Hendershott, T.; Jones, C.M.; Menkveld, A.J. Does Algorithmic Trading Improve Liquidity? J. Financ.
**2011**, 66, 1–33. [Google Scholar] [CrossRef] [Green Version] - Menkveld, A.J. High frequency trading and the new market makers. J. Financ. Mark.
**2013**, 16, 712–740. [Google Scholar] [CrossRef] [Green Version] - Chaboud, A.P.; Chiquoine, B.; Hjalmarsson, E.; Vega, C. Rise of the Machines: Algorithmic Trading in the Foreign Exchange Market. J. Financ.
**2014**, 69, 2045–2084. [Google Scholar] [CrossRef] [Green Version] - Dasar, M.; Sankar, S. Technology Systems in the Global FX Market | Celent. 2011. Available online: https://www.celent.com/insights/698289644 (accessed on 21 April 2020).
- Schmidt, A.B. Ecology of the Modern Institutional Spot FX: The EBS Market in 2011; Social Science Research Network: Rochester, NY, USA, 2012. [Google Scholar]
- Aite Group. Algo Trading Will Make Up 25% of FX Volume by 2014, Aite Says. Available online: https://www.euromoney.com/article/b12kjb9hbvjsj8/algo-trading-will-make-up-25-of-fx-volume-by-2014-aite-says (accessed on 21 April 2020).
- BIS, (Bank for International Settlements) Electronic Trading in Fixed Income Markets. Available online: https://www.bis.org/publ/mktc07.htm (accessed on 21 April 2020).
- Glantz, M.; Kissell, R. Chapter 8—Algorithmic Trading Risk. In Multi-Asset Risk Modeling; Glantz, M., Kissell, R., Eds.; Academic Press: San Diego, CA, 2014; pp. 247–304. [Google Scholar]
- Lee, S.; Tierney, D. The Trade Surveillance Compliance: Market and the Battle for Automation; Aite Group: Boston, MA, USA, 2013. [Google Scholar]
- Rennison, J.; Meyer, G.; Bullock, N. How high-frequency trading hit a speed bump. Financial Times. 2018. Available online: https://www.ft.com/content/d81f96ea-d43c-11e7-a303-9060cb1e5f44 (accessed on 21 April 2020).
- SEC, (U.S. Securities and Exchange Commission); BCG, (Boston Consulting Group). Organizational Study and Reform; Bethesda, MD, USA, 2011; Available online: https://www.sec.gov/news/studies/2011/967study.pdf (accessed on 21 April 2020).
- The Economist. The fast and the furious. The Economist. 25 February 2012. Available online: https://www.economist.com/special-report/2012/02/25/the-fast-and-the-furious (accessed on 21 April 2020).
- High-Frequency Trading in the Foreign Exchange Market. Available online: https://www.bis.org/publ/mktc05.pdf (accessed on 21 April 2020).
- Granham, P. Rise of Electronic FX Trading Won’t Silence Voice Brokers; Euromoney: London, UK, 2013; Available online: https://www.euromoney.com/article/b12kjx172lr4mn/rise-of-electronic-fx-trading-wont-silence-voice-brokers (accessed on 21 April 2020).
- Breedon, F.; Chen, L.; Ranaldo, A.; Vause, N. Judgement Day: Algorithmic Trading Around the Swiss Franc Cap Removal. Available online: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3126136 (accessed on 21 April 2020).
- Das, R.; Hanson, J.E.; Kephart, J.O.; Tesauro, G. Agent-human Interactions in the Continuous Double Auction. In Proceedings of the 17th International Joint Conference on Artificial Intelligence—Volume 2; Morgan Kaufmann Publishers Inc.: San Francisco, CA, USA, 2001; pp. 1169–1176. [Google Scholar]
- Chakravorty, G. Execution Algorithms: How Can We Get the Best Price for Our Trade? 2017. Available online: https://slides.com/gchak/execution-algorithms#/ (accessed on 21 April 2020).
- Jovanovic, B.; Menkveld, A.J. Middlemen in Limit Order Markets; Social Science Research Network: Rochester, NY, USA, 2016. [Google Scholar]
- Mishra, S.; Daigler, R.T.; Holowczak, R. The Effect of High-Frequency Market Making on Option Market Liquidity. J. Trading
**2016**, 11, 56–76. [Google Scholar] [CrossRef] - McGowan, M. The Rise of Computerized High Frequency Trading: Use and Controversy. Duke Law Technol. Rev.
**2010**, 9, 1–25. [Google Scholar] - Hendershott, T.; Moulton, P.C. Automation, speed, and stock market quality: The NYSE’s Hybrid. J. Financ. Mark.
**2011**, 14, 568–604. [Google Scholar] [CrossRef] - Simon, H.A. Theories of bounded rationality. Decis. Organ.
**1972**, 1, 161–176. [Google Scholar] - Wolpert, D.H. The Lack of A Priori Distinctions Between Learning Algorithms. Neural Comput.
**1996**, 8, 1341–1390. [Google Scholar] [CrossRef] - Wolpert, D.H.; Macready, W.G. No free lunch theorems for optimization. IEEE Trans. Evol. Comput.
**1997**, 1, 67–82. [Google Scholar] [CrossRef] [Green Version] - Tsvetkova, M.; García-Gavilanes, R.; Floridi, L.; Yasseri, T. Even good bots fight: The case of Wikipedia. PLoS ONE
**2017**, 12, e0171774. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Park, J.B.; Won Lee, J.; Yang, J.-S.; Jo, H.-H.; Moon, H.-T. Complexity analysis of the stock market. Physica A
**2007**, 379, 179–187. [Google Scholar] [CrossRef] [Green Version] - Ng, A. What Artificial Intelligence Can and Can’t Do Right Now. Harvard Business Review. 9 November 2016. Available online: https://hbr.org/2016/11/what-artificial-intelligence-can-and-cant-do-right-now (accessed on 21 April 2020).
- Hilbert, M. Big Data requires Big Visions for Big Change; TEDxUCL, x=independently organized TED talks: London, UK, 2014; Available online: https://www.martinhilbert.net/big-data-requires-big-visions-for-big-change-tedxucl/ (accessed on 21 April 2020).
- Madsen, A.K.; Flyverbom, M.; Hilbert, M.; Ruppert, E. Big Data: Issues for an International Political Sociology of Data Practices. Int. Political Sociol.
**2016**, olw010. [Google Scholar] [CrossRef] - Cvitanic, J.; Kirilenko, A.A. High Frequency Traders and Asset Prices; Social Science Research Network: Rochester, NY, USA, 2010. [Google Scholar]
- Dukascopy Bank. Historical Data Feed Swiss Forex Bank; Swiss FX trading platform; Dukascopy Bank SA: Geneva, Switzerland, 2018; Available online: https://www.dukascopy.com/swiss/english/marketwatch/historical/ (accessed on 21 April 2020).
- Smirnovs, J.; Hilbert, M. Personal email communication with Dukascopy representative, 2019.
- Marton, K.; Shields, P.C. Entropy and the Consistent Estimation of Joint Distributions. Ann. Probab.
**1994**, 22, 960–977. [Google Scholar] [CrossRef] - Lewis, M. Flash Boys: A Wall Street Revolt; W. W. Norton & Company: New York, NY, USA, 2014; ISBN 978-0-393-24467-0. [Google Scholar]
- Pappalardo, J. New Transatlantic Cable Built to Shave 5 Milliseconds off Stock Trades. Popular Mechanics. 27 October 2011. Available online: https://www.popularmechanics.com/technology/infrastructure/a7274/a-transatlantic-cable-to-shave-5-milliseconds-off-stock-trades/ (accessed on 21 April 2020).
- Cover, T.M.; Thomas, J.A. Elements of Information Theory, 2nd ed.; Wiley-Interscience: Hoboken, NJ, USA, 2006; ISBN 0-471-24195-4. [Google Scholar]
- Hilbert, M.; Darmon, D. Dumb and Predictable Bots Make Largescale Communication More Complex and Unpredictable. Available online: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3464833 (accessed on 21 April 2020).
- Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] [Green Version] - Kolmogorov, A.N. A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl. Akad. Nauk SSSR.
**1958**, 119, 2. [Google Scholar] - Kolmogorov, A.N. Entropy per unit time as a metric invariant of automorphisms. Dokl. Akad. Nauk SSSR
**1959**, 124, 754–755. [Google Scholar] - Sinai, Y.G. On the notion of entropy of dynamical systems. Dokl. Akad. Nauk
**1959**, 124, 768–771. [Google Scholar] - Crutchfield, J.P.; Feldman, D. Regularities unseen, randomness observed: Levels of entropy convergence. Chaos Interdiscip. J. Nonlinear Sci.
**2003**, 13, 25–54. [Google Scholar] [CrossRef] - Crutchfield, J.P.; Young, K. Inferring statistical complexity. Phys. Rev. Lett.
**1989**, 63, 105–108. [Google Scholar] [CrossRef] [PubMed] - Grassberger, P. Toward a quantitative theory of self-generated complexity. Int. J. Theor. Phys.
**1986**, 25, 907–938. [Google Scholar] [CrossRef] - Pincus, S.M. Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. USA
**1991**, 88, 2297–2301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Crutchfield, J.P.; Ellison, C.J.; Mahoney, J.R. Time’s Barbed Arrow: Irreversibility, Crypticity, and Stored Information. Phys. Rev. Lett.
**2009**, 103, 094101. [Google Scholar] [CrossRef] [Green Version] - James, R.G.; Ellison, C.J.; Crutchfield, J.P. Anatomy of a bit: Information in a time series observation. Chaos Interdiscip. J. Nonlinear Sci.
**2011**, 21, 037109. [Google Scholar] [CrossRef] [Green Version] - Yeung, R.W. A new outlook on Shannon’s information measures. IEEE Trans. Inf. Theory
**1991**, 37, 466–474. [Google Scholar] [CrossRef] - Crutchfield, J.P.; Feldman, D.P. Statistical Complexity of Simple 1D Spin Systems. Phys. Rev. E
**1997**, 55, R1239–R1242. [Google Scholar] [CrossRef] [Green Version] - Lindgren, K.; Nordahl, M.G. Complexity measures and cellular automata. Complex Syst.
**1988**, 2, 409–440. [Google Scholar] - Shaw, R. The Dripping Faucet as a Model Chaotic System; Aerial Press: Ann Arbor, MI, USA, 1984. [Google Scholar]
- Bialek, W.; Nemenman, I.; Tishby, N. Predictability, Complexity and Learning. Neural Comput.
**2001**, 13, 2001. [Google Scholar] [CrossRef] - Shalizi, C.R.; Crutchfield, J.P. Computational Mechanics: Pattern and Prediction, Structure and Simplicity. J. Stat. Phys.
**2001**, 104, 817–879. [Google Scholar] [CrossRef] - Crutchfield, J.P. Between order and chaos. Nat. Phys.
**2012**, 8, 17–24. [Google Scholar] [CrossRef] - Crutchfield, J.P. The calculi of emergence: Computation, dynamics and induction. Phys. D Nonlin. Phenomena
**1994**, 75, 11–54. [Google Scholar] [CrossRef] - Hilbert, M.; James, R.G.; Gil-Lopez, T.; Jiang, K.; Zhou, Y. The Complementary Importance of Static Structure and Temporal Dynamics in Teamwork Communication. Hum. Commun. Res.
**2018**, 44, 427–448. [Google Scholar] [CrossRef] - Darmon, D. Statistical Methods for Analyzing Time Series Data Drawn from Complex Social Systems. PhD Thesis, University of Maryland, College Park, MD, USA, 2015. Supervised by Michelle Girvan and William Rand. [Google Scholar] [CrossRef]
- Shalizi, C.R.; Klinkner, K.L. Blind Construction of Optimal Nonlinear Recursive Predictors for Discrete Sequences. arXiv
**2014**, arXiv:cs/0406011. [Google Scholar] - Crutchfield, J.P.; Ellison, C.J.; Riechers, P.M. Exact complexity: The spectral decomposition of intrinsic computation. Phys. Lett. A
**2016**, 380, 998–1002. [Google Scholar] [CrossRef] [Green Version] - James, R.G.; Ellison, C.J.; Crutchfield, J.P. dit: A Python package for discrete information theory. J. Open Source Softw.
**2018**, 3, 738. [Google Scholar] [CrossRef] - Beers, B. What indicators are used in exchange rate forecasting? Investopedia. 29 August 2018. Available online: https://www.investopedia.com/ask/answers/021715/what-economic-indicators-are-most-used-when-forecasting-exchange-rate.asp (accessed on 21 April 2020).
- TradingEconomics. TradingEconomics Database. Available online: https://tradingeconomics.com/ (accessed on 21 April 2020).
- Allison, P. Don’t Put Lagged Dependent Variables in Mixed Models. Statistical Horizons. 2 June 2015. Available online: https://statisticalhorizons.com/lagged-dependent-variables (accessed on 21 April 2020).
- Kahn, R.; Whited, T.M. Identification Is Not Causality, and Vice Versa. Rev. Corp. Financ. Stud.
**2018**, 7, 1–21. [Google Scholar] [CrossRef] - Koopmans, T.C. Identification Problems in Economic Model Construction. Econometrica
**1949**, 17, 125–144. [Google Scholar] [CrossRef] - Morgan, S.L.; Winship, C. Counterfactuals and Causal Inference: Methods and Principles for Social Research. Available online: /core/books/counterfactuals-and-causal-inference/5CC81E6DF63C5E5A8B88F79D45E1D1B7 (accessed on 3 April 2020).
- Pearl, J. Causality; Cambridge University Press: Cambridge, UK, 2009; ISBN 978-0-521-89560-6. [Google Scholar]
- Wooldridge, J.M. Introductory Econometrics: A Modern Approach, 4th ed.; South-Western: Mason, OH, USA, 2008; ISBN 978-0-324-66054-8. [Google Scholar]
- Feldman, D.P.; McTague, C.S.; Crutchfield, J.P. The organization of intrinsic computation: Complexity-entropy diagrams and the diversity of natural information processing. Chaos
**2008**, 18, 043106. [Google Scholar] [CrossRef] - Osborne, M.F.M. Periodic Structure in the Brownian Motion of Stock Prices. Oper. Res.
**1962**, 10, 345–379. [Google Scholar] [CrossRef] - Niederhoffer, V. Clustering of Stock Prices. Oper. Res.
**1965**, 13, 258–265. [Google Scholar] [CrossRef] - Harris, L. Stock Price Clustering and Discreteness. Rev. Financ. Stud.
**1991**, 4, 389–415. [Google Scholar] [CrossRef] - Tseng, M.; Mahmoodzadeh, S.; Gencay, R. Impact of Algorithmic Trading on Market Quality: A Reconciliation; Social Science Research Network: Rochester, NY, USA, 2018. [Google Scholar]
- Lallouache, M.; Abergel, F. Tick size reduction and price clustering in a FX order book. Physica A
**2014**, 416, 488–498. [Google Scholar] [CrossRef] - Ball, C.A.; Torous, W.N.; Tschoegl, A.E. The degree of price resolution: The case of the gold market. J. Futures Mark.
**1985**, 5, 29–43. [Google Scholar] [CrossRef] - Cooney, J.W.; Van Ness, B.; Van Ness, R. Do investors prefer even-eighth prices? Evidence from NYSE limit orders. J. Bank. Financ.
**2003**, 27, 719–748. [Google Scholar] [CrossRef] - Christie, W.G.; Harris, J.H.; Schultz, P.H. Why did NASDAQ Market Makers Stop Avoiding Odd-Eighth Quotes? J. Financ.
**1994**, 49, 1841–1860. [Google Scholar] [CrossRef] - Christie, W.G.; Schultz, P.H. Why do NASDAQ Market Makers Avoid Odd-Eighth Quotes? J. Financ.
**1994**, 49, 1813–1840. [Google Scholar] [CrossRef] - Mahmoodzadeh, S.; Gençay, R. Human vs. high-frequency traders, penny jumping, and tick size. J. Bank. Financ.
**2017**, 85, 69–82. [Google Scholar] [CrossRef] - Feynman, R. There’s Plenty of Room at the Bottom: An Invitation to Enter a New Field of Physics. Available online: https://www.taylorfrancis.com/books/e/9781315217178/chapters/10.1201/9781315217178-6 (accessed on 21 April 2020).
- Crutchfield, J.P. The Origins of Computational Mechanics: A Brief Intellectual History and Several Clarifications. arXiv
**2017**, arXiv:1710.06832. [Google Scholar] - Crutchfield, J.P.; Packard, N.H. Symbolic dynamics of noisy chaos. Physica D
**1983**, 7, 201–223. [Google Scholar] [CrossRef] - Seth, A.K.; Dienes, Z.; Cleeremans, A.; Overgaard, M.; Pessoa, L. Measuring consciousness: Relating behavioural and neurophysiological approaches. Trends Cognit. Sci.
**2008**, 12, 314–321. [Google Scholar] [CrossRef] [PubMed] - Seth, A.K.; Izhikevich, E.; Reeke, G.N.; Edelman, G.M. Theories and measures of consciousness: An extended framework. Proc. Natl. Acad. Sci. USA
**2006**, 103, 10799–10804. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sipser, M. Introduction to the Theory of Computation, 2nd ed.; Course Technology: Boston, MA, USA, 2006; ISBN 978-0-534-95097-2. [Google Scholar]
- Schreiber, T. Measuring Information Transfer. Phys. Rev. Lett.
**2000**, 85, 461–464. [Google Scholar] [CrossRef] [Green Version] - Amblard, P.-O.; Michel, O.J.J. On directed information theory and Granger causality graphs. J. Comput. Neurosci.
**2011**, 30, 7–16. [Google Scholar] [CrossRef] [PubMed] - Barnett, L.; Barrett, A.B.; Seth, A.K. Granger Causality and Transfer Entropy Are Equivalent for Gaussian Variables. Phys. Rev. Lett.
**2009**, 103, 238701. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Olbrich, E.; Bertschinger, N.; Rauh, J. Information Decomposition and Synergy. Entropy
**2015**, 17, 3501–3517. [Google Scholar] [CrossRef] [Green Version] - Williams, P.L.; Beer, R.D. Nonnegative Decomposition of Multivariate Information. arXiv
**2010**, arXiv:1004.2515. [Google Scholar] - Barnett, N.; Crutchfield, J.P. Computational Mechanics of Input–Output Processes: Structured Transformations and the \epsilon-Transducer. J. Stat. Phys.
**2015**, 161, 404–451. [Google Scholar] [CrossRef] [Green Version] - Cartlidge, J.; Szostek, C.; Luca, M.D.; Cliff, D. Too Fast Too Furious—Faster Financial-market Trading Agents Can Give Less Efficient Markets. In Proceedings of the 4th ICAART, Vilamoura, Portugal, 6–8 February 2012. [Google Scholar]

**Figure 2.**(

**a**) Info-diagram of entropies of past (H(past)) and future (H(future)), relating predictive complexity (C), predictable information (E), and remaining uncertainty (h * L = H(future|past)); (

**b**) schematic illustration of how to obtain statistics from a sequence consisting of a categorical variable with four different bins (A, B, C, D) by employing a sliding window of length L = 3.

**Figure 4.**Regression coefficients for bi-monthly changes in complexity in form of predictable information E

_{eM}(

**a**,

**c**) and predictive complexity C (

**b**,

**d**), where algorithmic trading is measured in 20 coarse-grained bins (

**a**,

**b**) and 200 fine-grained bins (

**c**,

**d**), indicating 95% confidence intervals with error bars. *** p < 0.01, ** p < 0.05, * p < 0.1 (N = 520).

**Figure 5.**Regression coefficients for bi-monthly changes in remaining uncertainty in form of entropy rate h

_{eM}, where algorithmic trading is measured in 20 coarse-grained bins (

**a**) and 200 fine-grained bins (

**b**), indicating 95% confidence intervals with error bars. *** p < 0.01, ** p < 0.05, * p < 0.1 (N = 520).

**Figure 6.**Estimated marginal means (large dots) from ANVOCA (95% confidence error bars), with thirds of AT

_{emp}as fixed factor, controlling for the covariates dep

_{t−1}, GDPr, infl, intr, unpl; orange circles: third with lowest algorithmic trading (AT

_{low}); blue triangles: third with highest algorithmic trading (AT

_{high}); (

**a**–

**c**) for coarse-grained perspective of 20 bins, (

**d**–

**f**) for fine-grained perspective of 200 bins. Note: in contrary to means, scatter data points are not corrected for control variables and some outlier are cut off by presentation.

**Figure 7.**Frequency distribution of bid-ask spreads EUR/AUD (

**a**) Jan.–Feb. 2007, unbinned and in 20 bins; (

**b**) Nov.–Dec. 2017, unbinned and in 20 bins; (

**c**) unbinned both 2007 and 2017, logarithmic scale.

**Figure 8.**Chain rule of entropy applied to EUR/AUD bid-ask spreads, with 20 and 200 bins. (

**a**) visualizes the diverging tendency over time; (

**b**) isolates the size of this area.

**Figure 9.**Visualization of ϵ-machines (aka predictive state machines) derived for (

**a**) AUD/JPY Mar.–Apr. 2009, 200 bins; (

**b**) EUR/AUD Jan.–Feb. 2017, 200 bins. The size of a node represents the steady state probability of the corresponding state. Clockwise curve indicates transition directionality. Transition color corresponds to symbols in the alphabet in the sequence.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hilbert, M.; Darmon, D.
How Complexity and Uncertainty Grew with Algorithmic Trading. *Entropy* **2020**, *22*, 499.
https://doi.org/10.3390/e22050499

**AMA Style**

Hilbert M, Darmon D.
How Complexity and Uncertainty Grew with Algorithmic Trading. *Entropy*. 2020; 22(5):499.
https://doi.org/10.3390/e22050499

**Chicago/Turabian Style**

Hilbert, Martin, and David Darmon.
2020. "How Complexity and Uncertainty Grew with Algorithmic Trading" *Entropy* 22, no. 5: 499.
https://doi.org/10.3390/e22050499