# Exploring the Quotation Inertia in International Currency Markets

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Computational Experiments Design

_{0}) ± dL is satisfied. Here, Y(t

_{0}) represents the value of the process at the time t

_{0}when the trend is detected, and dC denotes the level at which the trend is confirmed. For simplicity, we use $dL$ as both the detection and confirmation level, i.e., $dC=dL$.

#### 3.1. Experiment 1: Initial Process

**u**is determined via the Laplace function value $\mathrm{\u0424}\left({\mathrm{u}}_{cr}\right)=\frac{1}{2}-\frac{\mathsf{\alpha}}{2}=\frac{1-\mathsf{\alpha}}{2}=0.005$, where $\mathsf{\alpha}=0.99$.

_{cr}#### 3.2. Experiment 2: Smoothed Process with Reduced Segmentation

#### 3.3. Experiment 3: Smoothed Opening Process

#### 3.4. Experiment 4: Smoothed Opening and Closing Process

#### 3.5. Experiment 5: Comparison of Various Computational Schemes

#### 3.6. Experiment 6: Smoothed System Component of a Stochastic Process

#### 3.7. Experiment 7: Various Levels of Trend Confirmation

#### Discussion of Analysis of Inertia of Stochastic Processes Method Based on Qualitative Characteristics of Local Trends

^{*}, and trend confirmation level $dL$.

#### 3.8. Experiment 8

#### 3.9. Experiment 9

#### 3.10. Experiment 10

#### Discussion on the Analysis of Inertia in Stochastic Processes with Quantitative and Qualitative Characteristics of Local Trends

#### 3.11. Experiment 11

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Sciama, D.W. On the origin of inertia. Mon. Not. R. Astron. Soc.
**1953**, 113, 34–42. [Google Scholar] [CrossRef] - Cain, B.E. Inertia theory. Linear Algebra Its Appl.
**1980**, 30, 211–240. [Google Scholar] [CrossRef] - Datta, B.N. Stability and inertia. Linear Algebra Its Appl.
**1999**, 302, 563–600. [Google Scholar] [CrossRef] - Greve, H.R. The effect of core change on performance: Inertia and regression toward the mean. Adm. Sci. Q.
**1999**, 44, 590–614. [Google Scholar] [CrossRef] - Illeditsch, P.K. Ambiguous information, portfolio inertia, and excess volatility. J. Financ.
**2011**, 66, 2213–2247. [Google Scholar] [CrossRef] - Peters, E.E. Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market Volatility, 2nd ed.; John Wiley & Sons: New York, NY, USA, 1996. [Google Scholar]
- Pan, H.; Long, M. Intelligent portfolio theory and application in stock investment with multi-factor models and trend following trading strategies. Procedia Comput. Sci.
**2021**, 187, 414–419. [Google Scholar] [CrossRef] - Bukunov, S.V. Verification of the applicability of trend strategies in modern financial markets. In Proceedings of the International Science and Technology Conference “FarEastCon 2020”, Vladivostok, Russia, 6–9 October 2020; pp. 879–886. [Google Scholar]
- Hurst, B.; Ooi, Y.H.; Pedersen, L.H. A century of evidence on trend-following investing. J. Portf. Manag.
**2017**, 44, 15–29. [Google Scholar] [CrossRef] - Gregory-Williams, J.; Williams, B.M. Trading Chaos: Maximize Profits with Proven Technical Techniques, 2nd ed.; John Wiley & Sons: New York, NY, USA, 2004. [Google Scholar]
- LeBaron, B. Chaos and nonlinear forecastability in economics and finance. Philos. Trans. R. Soc. London. Ser. A Phys. Eng. Sci.
**1994**, 348, 397–404. [Google Scholar] - Bucolo, M.; Buscarino, A.; Famoso, C.; Fortuna, L.; Frasca, M. Control of imperfect dynamical systems. Nonlinear Dyn.
**2019**, 98, 2989–2999. [Google Scholar] [CrossRef] - Rafael’M, Y.; Musaev, A.A.; Grigoriev, D.A. Evaluation of statistical forecast method efficiency in the conditions of dynamic chaos. In Proceedings of the 2021 IV International Conference on Control in Technical Systems (CTS), Saint Petersburg, Russia, 21–23 September 2021; pp. 178–180. [Google Scholar]
- Musaev, A.; Makshanov, A.; Grigoriev, D. Forecasting multivariate chaotic processes with precedent analysis. Computation
**2021**, 9, 110. [Google Scholar] [CrossRef] - Musaev, A.; Grigoriev, D. Numerical Studies of Statistical Management Decisions in Conditions of Stochastic Chaos. Mathematics
**2022**, 10, 226. [Google Scholar] [CrossRef] - Bolch, B.W.; Huang, C.J. Multivariate Statistical Methods for Business and Economics; Prentice Hall: Hoboken, NJ, USA, 1974. [Google Scholar]
- Chihara, L.M.; Hesterberg, T.C. Mathematical Statistics with Resampling and R, 2nd ed.; John Wiley & Sons, Ltd.: New York, NY, USA, 2018. [Google Scholar]
- Kendall, M.G.; Stuart, A. The Advanced Theory of Statistics, 2nd ed.; Charles Griffin & Co., Ltd.: London, UK, 1963; Volume 2. [Google Scholar]
- Shao, J. Mathematical Statistics; Springer Science & Business Media: New York, NY, USA, 2003. [Google Scholar]
- Sauermann, J. On the instability of majority decision-making: Testing the implications of the ‘chaos theorems’ in a laboratory experiment. Theory Decis.
**2020**, 88, 505–526. [Google Scholar] [CrossRef] - Masson, M.E. A tutorial on a practical Bayesian alternative to null-hypothesis significance testing. Behav. Res. Methods
**2011**, 43, 679–690. [Google Scholar] [CrossRef] [PubMed] - Everette, S.G., Jr. Exponential smoothing: The state of the art. J. Forecast.
**1985**, 4, 1–28. [Google Scholar] - Musaev, A.; Makshanov, A.; Grigoriev, D. Numerical studies of channel management strategies for nonstationary immersion environments: EURUSD case study. Mathematics
**2022**, 10, 1408. [Google Scholar] [CrossRef]

**Figure 5.**An example of an analysis of a simple strategy based on the dynamics of linear trends over an interval of 10 days.

Time Interval, Days | Currency Instruments | ||
---|---|---|---|

EURUSD | EURJPY | USDJPY | |

1–100 | 0.552 | 0.484 | 0.444 |

101–200 | 0.507 | 0.536 | 0.465 |

201–300 | 0.533 | 0.552 | 0.560 |

301–400 | 0.494 | 0.452 | 0.465 |

401–500 | 0.446 | 0.545 | 0.444 |

Time Interval, Days | Currency Instruments | ||
---|---|---|---|

EURUSD | EURJPY | USDJPY | |

1–100 | 0.539 | 0.568 | 0.522 |

101–200 | 0.524 | 0.528 | 0.497 |

201–300 | 0.529 | 0.503 | 0.537 |

301–400 | 0.503 | 0.550 | 0.534 |

401–500 | 0.493 | 0.548 | 0.552 |

**Table 3.**The frequency of positive outcomes for positions that were opened using the smoothed process.

Time Interval, Days | EURUSD | ||
---|---|---|---|

$\mathsf{\alpha}=0.005$ | $\mathsf{\alpha}=0.01$ | $\mathsf{\alpha}=0.02$ | |

1–100 | 0.667 | 0.681 | 0.618 |

101–200 | 0.771 | 0.791 | 0.667 |

201–300 | 0.606 | 0.706 | 0.612 |

301–400 | 0.612 | 0.653 | 0.618 |

401–500 | 0.648 | 0.581 | 0.574 |

Time Interval, Days | EURUSD | ||
---|---|---|---|

$\mathsf{\alpha}=0.005$ | $\mathsf{\alpha}=0.01$ | $\mathsf{\alpha}=0.02$ | |

1–100 | 0.652 | 0.652 | 0.593 |

101–200 | 0.698 | 0.706 | 0.696 |

201–300 | 0.686 | 0.707 | 0.688 |

301–400 | 0.612 | 0.612 | 0.582 |

401–500 | 0.567 | 0.534 | 0.574 |

Time Interval, Days | EURUSD | EURJPY | ||||
---|---|---|---|---|---|---|

Scheme 1 | Scheme 2 | Scheme 3 | Scheme 4 | Scheme 5 | Scheme 6 | |

1–100 | 0.54 | 0.52 | 0.71 | 0.49 | 0.52 | 0.67 |

101–200 | 0.50 | 0.53 | 0.72 | 0.50 | 0.54 | 0.73 |

201–300 | 0.54 | 0.50 | 0.72 | 0.53 | 0.54 | 0.72 |

301–400 | 0.48 | 0.43 | 0.69 | 0.50 | 0.50 | 0.69 |

401–500 | 0.48 | 0.43 | 0.66 | 0.51 | 0.55 | 0.72 |

Time Interval, Days | EURUSD, dL = 75 | EURJPY, dL = 75 | ||||

$\mathsf{\alpha}$ = 0.02 | 0.01 | 0.005 | 0.02 | 0.01 | 0.005 | |

1–100 | 0.69 | 0.61 | 0.71 | 0.57 | 0.62 | 0.68 |

101–200 | 0.72 | 0.76 | 0.75 | 0.72 | 0.74 | 0.81 |

201–300 | 0.66 | 0.66 | 0.65 | 0.70 | 0.73 | 0.78 |

301–400 | 0.55 | 0.57 | 0.60 | 0.67 | 0.70 | 0.79 |

401–500 | 0.58 | 0.59 | 0.63 | 0.68 | 0.71 | 0.79 |

Time Interval,Days | EURUSD, dL = 50 | EURJPY, dL = 50 | ||||

$\mathsf{\alpha}$ = 0.02 | 0.01 | 0.005 | 0.02 | 0.01 | 0.005 | |

1–100 | 0.58 | 0.63 | 0.70 | 0.63 | 0.70 | 0.71 |

101–200 | 0.70 | 0.72 | 0.80 | 0.75 | 0.76 | 0.82 |

201–300 | 0.71 | 0.76 | 0.78 | 0.72 | 0.76 | 0.80 |

301–400 | 0.66 | 0.67 | 0.72 | 0.70 | 0.71 | 0.75 |

401–500 | 0.65 | 0.67 | 0.68 | 0.70 | 0.73 | 0.77 |

Time Interval, Days | EURUSD, dL = 100 | EURJPY, dL = 100 | ||||
---|---|---|---|---|---|---|

dC = 75 | dC = 50 | dC =25 | dC = 75 | dC = 50 | dC = 25 | |

1–100 | 0.63 | 0.67 | 0.74 | 0.64 | 0.79 | 0.75 |

101–200 | 0.75 | 0.75 | 0.90 | 0.75 | 0.79 | 0.84 |

201–300 | 0.72 | 0.73 | 0.79 | 0.66 | 0.65 | 0.84 |

301–400 | 0.62 | 0.63 | 0.76 | 0.55 | 0.68 | 0.81 |

401–500 | 0.60 | 0.61 | 0.72 | 0.59 | 0.63 | 0.83 |

Time Interval, Days | EURUSD, dL = 75 | EURUSD, dL = 50 | ||||
---|---|---|---|---|---|---|

dC = 75 | dC = 50 | dC = 25 | dC = 75 | dC = 50 | dC = 25 | |

0–100 | 0.57 | 0.70 | 0.74 | 0.67 | 0.68 | 0.71 |

101–200 | 0.72 | 0.73 | 0.84 | 0.75 | 0.75 | 0.84 |

201–300 | 0.70 | 0.74 | 0.78 | 0.69 | 0.72 | 0.80 |

301–400 | 0.67 | 0.69 | 0.80 | 0.68 | 0.70 | 0.80 |

401–500 | 0.68 | 0.71 | 0.79 | 0.66 | 0.70 | 0.78 |

${\mathit{a}}^{\mathbf{*}}$ | $\mathit{d}\mathit{L},\mathit{p}.\backslash \mathit{l},\mathbf{D}\mathbf{a}\mathbf{y}\mathbf{s}$ | 0.025 | 0.05 | 0.075 | 0.1 |
---|---|---|---|---|---|

0.025 | 25 | 0.50 | 0.48 | 0.49 | 0.49 |

0.025 | 50 | 0.51 | 0.50 | 0.51 | 0.50 |

0.025 | 75 | 0.50 | 0.50 | 0.50 | 0.51 |

0.025 | 100 | 0.50 | 0.51 | 0.51 | 0.51 |

0.05 | 25 | 0.50 | 0.48 | 0.48 | 0.50 |

0.05 | 50 | 0.50 | 0.50 | 0.51 | 0.50 |

0.05 | 75 | 0.50 | 0.50 | 0.51 | 0.50 |

0.05 | 100 | 0.50 | 0.51 | 0.51 | 0.51 |

0.075 | 25 | 0.50 | 0.49 | 0.49 | 0.49 |

0.075 | 50 | 0.50 | 0.50 | 0.50 | 0.50 |

0.075 | 75 | 0.50 | 0.50 | 0.51 | 0.51 |

0.075 | 100 | 0.51 | 0.51 | 0.51 | 0.51 |

0.1 | 25 | 0.50 | 0.48 | 0.48 | 0.49 |

0.1 | 50 | 0.50 | 0.51 | 0.51 | 0.50 |

0.1 | 75 | 0.50 | 0.50 | 0.50 | 0.50 |

0.1 | 100 | 0.50 | 0.50 | 0.50 | 0.51 |

Frequency of Trend Confirmation | ||||
---|---|---|---|---|

$\mathit{d}\mathit{L},\mathbf{p}.\backslash \mathsf{\Delta}\mathit{T}$ | Days 1–100 | Days 101–200 | Days 201–300 | Days 301–400 |

25 | 0.48 | 0.49 | 0.48 | 0.47 |

50 | 0.53 | 0.46 | 0.48 | 0.48 |

75 | 0.53 | 0.50 | 0.49 | 0.45 |

100 | 0.55 | 0.50 | 0.52 | 0.47 |

Frequency of Trend Confirmation | ||||
---|---|---|---|---|

w, min | Days 1–100 | Days 101–200 | Days 201–300 | Days 301–400 |

100 | 0.54 | 0.46 | 0.54 | 0.54 |

200 | 0.48 | 0.51 | 0.50 | 0.50 |

300 | 0.49 | 0.48 | 0.52 | 0.52 |

Groups | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

dC | 75 | 100 | 50 | 75 | 75 | 75 | 75 |

dL | 75 | 75 | 75 | 100 | 50 | 75 | 75 |

dT | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1500 |

**Table 13.**Frequency of inertia confirmation for various time intervals of observation for the initial data.

Observation Interval Parameter Group | 1–100 | 101–200 | 201–300 | 301–400 | 401–500 |
---|---|---|---|---|---|

1 | 0.51 | 0.54 | 0.46 | 0.47 | 0.52 |

2 | 0.36 | 0.54 | 0.54 | 0.47 | 0.50 |

3 | 0.49 | 0.45 | 0.43 | 0.49 | 0.49 |

4 | 0.57 | 0.59 | 0.46 | 0.55 | 0.50 |

5 | 0.58 | 0.51 | 0.46 | 0.42 | 0.53 |

6 | 0.61 | 0.81 | 0.84 | 0.41 | 0.43 |

7 | 0.56 | 0.55 | 0.47 | 0.45 | 0.48 |

**Table 14.**Frequency of inertia confirmation for a smoothed stochastic process and various observation intervals using the smoothed component only to detect the trend.

Observation Interval Parameter Group | 1–100 | 101–200 | 201–300 | 301–400 | 401–500 |
---|---|---|---|---|---|

1 | 0.49 | 0.50 | 0.52 | 0.46 | 0.31 |

2 | 0.46 | 0.46 | 0.48 | 0.45 | 0.57 |

3 | 0.45 | 0.46 | 0.55 | 0.50 | 0.47 |

4 | 0.41 | 0.52 | 0.51 | 0.48 | 0.50 |

5 | 0.49 | 0.48 | 0.45 | 0.47 | 0.35 |

6 | 0.49 | 0.46 | 0.51 | 0.44 | 0.40 |

7 | 0.75 | 0.49 | 0.48 | 0.69 | 0.44 |

**Table 15.**Frequency of inertia confirmation for a smoothed stochastic process and various observation intervals using the smoothed component to detect and confirm the trend.

Observation Interval Parameter Group | 1–100 | 101–200 | 201–300 | 301–400 | 401–500 |
---|---|---|---|---|---|

1 | 0.55 | 0.82 | 0.85 | 0.46 | 0.44 |

2 | 0.22 | 0.68 | 0.88 | 0.27 | 0.43 |

3 | 0.69 | 0.81 | 0.83 | 0.54 | 0.39 |

4 | 0.47 | 0.72 | 0.77 | 0.24 | 0.31 |

5 | 0.55 | 0.76 | 0.87 | 0.54 | 0.33 |

6 | 0.61 | 0.81 | 0.84 | 0.45 | 0.44 |

7 | 0.56 | 0.86 | 0.90 | 0.45 | 0.71 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Musaev, A.; Makshanov, A.; Grigoriev, D.
Exploring the Quotation Inertia in International Currency Markets. *Computation* **2023**, *11*, 209.
https://doi.org/10.3390/computation11110209

**AMA Style**

Musaev A, Makshanov A, Grigoriev D.
Exploring the Quotation Inertia in International Currency Markets. *Computation*. 2023; 11(11):209.
https://doi.org/10.3390/computation11110209

**Chicago/Turabian Style**

Musaev, Alexander, Andrey Makshanov, and Dmitry Grigoriev.
2023. "Exploring the Quotation Inertia in International Currency Markets" *Computation* 11, no. 11: 209.
https://doi.org/10.3390/computation11110209