# Statistical Analysis of Current Financial Instrument Quotes in the Conditions of Market Chaos

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## Abstract

**:**

## 1. Introduction

## 2. Analyzed Data and Methodology

#### 2.1. Data Model

_{k}= x

_{k}+ v

_{k}, k = 1, …, n,

_{k}, k = 1, …, n is a smoothed system component used for constructing management strategies and v

_{k}, k = 1, …, n is noise. Similarly, instead of y

_{k,}first finite differences of their logarithms could be used (i.e., the ARCH or GARCH models) [11,12], but this complicates returning to an estimation of the initial y

_{k}. Let us highlight two important features of the presented observation model:

- the system component is comprised of an oscillatory non-periodic observation series typical for non-linear dynamics processes; and

_{k}= αy

_{k}+ (1 − α)y

_{(k−1)}= x

_{(k−1)}+ α(y

_{k}− x

_{(k−1)}), k = 2, …, n

_{k,}k = 1, …, n does not conform to the efficient market hypothesis [7] and, as shown in [16,17], has almost no inertia. The latter statement leads to a complete failure of management strategies based on mechanistic prolongation of the detected trends. At the same time, useful patterns can also be found in the system of correlations of trading asset quotations, as they are represented by multivariate observation series. In particular, increasing the observation interval makes it possible to detect stable correlations between various instruments of the currency and other markets [18,19].

#### 2.2. Problem Statement

_{Rk}= X

_{Rk}+ V

_{Rk},

where Y

_{Rk}= (y

_{R1}, y

_{R2}, …, y

_{Rm})

_{k}, X

_{Rk}= (x

_{R1}, x

_{R2}, …, x

_{Rm})

_{k},

V

_{Rk}= (v

_{R1}, v

_{R2}, …, v

_{Rm}), k = 1, …, n.

_{k}, k = 1, …, n. In the future, the elements Y

_{R}will be used as regressors in the traditional MR model

_{k}= C

_{k}Y

_{Rk}+ v

_{k}, k = 1, …, n

_{k}, k = 1, …, n, as already noted, is via the use of a conventional computational scheme based on the LSM:

_{k}= (Y

_{R[1:k−1,m]}

^{T}Y

_{R[1:k−1,m]})

^{−1}Y

_{R[1:k−1,m]}

^{T}Y

_{[1:k−1, 1]}, k = 2, …, n

where Y

_{R[1:k−1,m]}= (Y

_{R,1}, …, Y

_{R,k−1}), Y

_{[1:k−1,1]}= (y

_{1}, …, y

_{k−1}), k = 2, …, m

_{k,}k = 1, …, n and its regression estimate (4), reflecting the opinion of the market, represented by a set of regressor instruments, about its real value. If the regressors are representative in terms of their ability to reflect significant market variations, then the difference

_{k}= ŷ

_{k}− y

_{k}, k = 1, …, n

#### 2.3. A Simple Management Strategy Based on Multiregression Estimation

_{k}| > d*, k = 1, …, n. If d

_{k}> d*, this means that ŷ

_{k}> y + d*, i.e., the financial instrument is underpriced, and its price can be expected to go up. Vice versa, d

_{k}< −d* indicates an overprice, and therefore the instrument’s price should be expected to go down.

_{k}estimated on the selected observation interval. The corresponding estimate of the SD of the difference process was s = 24.7 and s = 21.5 for the smoothed process.

_{k}, k = 1, …, n, which demonstrates the weak convergence of the given difference’s distribution to the Gaussian law. This picture is static; considering it dynamics-wise, it can be seen that all the moments of the distribution change over time, and therefore the process is purely non-stationary.

_{k}, k = 1, …, n (Figure 7). For easier visualization, a relatively small observation interval (5 days) is considered, and the oscillator and its critical value are enlarged by 1.5 times.

_{k}> d*, a long position should be opened. Alternatively, one can open a short position if d

_{k}< −d*. Positions can be closed along with a reverse crossing of the threshold d*, or by the conventional methods of setting Take Profit and Stop Loss levels.

#### 2.4. Specifics of Financial Instrument Value Analysis in the Conditions of Non-Stationary and Non-Linear Dynamics

- The system component of a series of observations (1) x
_{k}, k = 1, …, n is an unknown deterministic process, which in some cases can be expressed analytically; - Regressors are not mutually correlated, i.e.,cov(y
_{Ri}, y_{rj}) = 0, ∀i ≠ j, i, j = 1, …, m; and - The noise component of the model (1) v
_{k}, k = 1, …, n is stationary, centered relative to the system component, and uncorrelated to the regressor’s random processE{v_{k}} = 0, cov{v_{i},v_{j}}_{k}= 0; ∀i ≠ j;

E{v_{k}, y_{Ri,k}} = 0, i = 1, … ,m, k = 1, …, n.

_{R[k−L:k−1,m]}= (Y

_{R,k−L}, …, Y

_{R,k−1}), Y

_{[k−L:k−1,1]}= (y

_{k−L}, …, y

_{k−1}), k = 1, …, m

^{2}= s

^{2}+ b

^{2}= min(L), where s is the estimate of the SD, and b is the estimated bias. However, for a model of the form (1) with a chaotic system component, it is not possible to obtain stable estimates of SD and bias values. In this regard, the choice of the observation window is carried out empirically, by comparing the estimation results on the training observation window preceding the current time.

_{Rk}, k = 1, …, n in the computational scheme (4) will be different at each time moment. This justifies the use of structural adaptation of the MR model with a step-by-step selection of financial instruments used as regressors.

_{k}, k = 1, …, n. At the same time, this approach inevitably leads to delays in the reaction of current estimates to significant changes in the relative dynamics of quotations of financial instruments. In other words, there is a contradiction between the quality of smoothing random fluctuations in observations and the growth of the estimation bias caused by dynamic errors. An illustration of this problem is given by the example of the allocation of the system component of the quotes of the EURUSD currency pair using the exponential filter (2) for the filter coefficients $\mathsf{\alpha}=0.1$ (Figure 9a) and $\mathsf{\alpha}=0.01$ (Figure 9b).

## 3. Results

_{k}(top) on a three-day observation interval. The same figure shows an example of the implementation of a simple management strategy based on an adaptive MR estimate. If the difference d

_{k}between the estimate and the current value of a currency instrument turns out to be greater (in absolute value) than the threshold value d*, a recommendation is made to open a position in the appropriate direction. The asterisk marks the state of the quote at the time of opening a long position. The diamond corresponds to the position closing at d

_{k}intersecting back the threshold value d*.

_{k}, k = 1, …, n of the threshold value d* is crossed. The position is closed when the process d

_{k+τ}crosses the level d*

_{k+τ}back or the zero level d

_{k+τ}= 0, where τ is the time of the crossing.

_{a}is formed. Further, in a loop over the number of generations N

_{g}, a new generation is created, consisting of an already existing group of ancestral genomes and a newly generated group of descendant genomes (DG). Descendant genomes are constructed from ancestral genomes in three main ways, including:

- Small single changes made to one of the AG parameters. The parameter in question is selected by a random draw. If changes are to be made sequentially to each parameter, then each AG receives m
_{g}modifications, where m_{g}is the size of the genome. In this case, there are N^{(1)}_{d}= N_{a}m_{g}descendants with a given type of modification, and only one parameter (gene) is modified in each of them. In this case, m_{g}= 3; therefore, if N_{a}= 4 best variants (ancestors) are preserved in each generation, N^{(1)}_{d}= 12 versions of the first type of DG will be obtained. - Small group changes. They are carried out similarly to the previous case, but are made to all parameters at once instead. Thus, there are N
^{(2)}_{d}= 4 more versions of DGs with slow changes in all genes. - Strong single mutation or parametric mutation. The AG and the gene number are selected via a random draw. With the probability of parametric mutation, ${P}_{pm}$ produces N
^{(3)}_{d}descendants, in each of which one gene in the range |∆| > 3σ is modified.

_{gc}= 9 generation changes in the same time interval of 10 days. As a starting genome, we used a vector G

_{0}= [nW

_{0}, α

_{0}, d*

_{0}] = [5, 0.01, 0.6]. During genome modification, we used rough estimates of the SD of the three listed parameters: SD(G) = [3, 0.02, 0.5].

- A fixed group of five regressors is selected before the start of trading operations maximizing the correlation with the working instrument (asset). The correlation matrix for all 16 financial instruments is evaluated based on observations of their quotes during the 15 days preceding the start of trading.
- Sequential adjustment of a group of five regressors. The adjustment is carried out based on the maximum correlation with the working instrument (asset) with an interval of 10 h. The correlation matrix is evaluated based on the results of observations of their quotes on a sliding observation window of 15 days.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Panayiotis, T. Oil volatility index and Chinese stock markets during financial crisis: A time-varying perspective. J. Chin. Econ. Foreign Trade Stud.
**2021**, 14, 187–201. [Google Scholar] - Lee, R. Quantum Trader—A Multiagent-Based Quantum Financial Forecast and Trading System. In Quantum Finance; Springer: Singapore, 2020; pp. 375–398. [Google Scholar]
- Lee, R. COSMOS trader–Chaotic Neuro-oscillatory multiagent financial prediction and trading system. J. Financ. Data Sci.
**2019**, 5, 61–82. [Google Scholar] [CrossRef] - Marti, G.; Nielsen, F.; Binkowski, M.; Donnat, P. A review of two decades of correlations, hierarchies, networks and clustering in financial markets. arXiv
**2017**, arXiv:1703.00485. [Google Scholar] - Moews, B.; Herrmann, J.M.; Ibikunle, G. Lagged correlation-based deep learning for directional trend change prediction in financial time series. Expert Syst. Appl.
**2019**, 120, 197–206. [Google Scholar] [CrossRef] [Green Version] - Wątorek, M.; Drożdż, S.; Oświȩcimka, P.; Stanuszek, M. Multifractal cross-correlations between the world oil and other financial markets in 2012–2017. Energy Econ.
**2019**, 81, 874–885. [Google Scholar] [CrossRef] [Green Version] - 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; 288p. [Google Scholar]
- Gregory-Williams, J.; Williams, B.M. Trading Chaos: Maximize Profits with Proven Technical Techniques, 2nd ed.; John Wiley & Sons: New York, NY, USA, 2004; 251p. [Google Scholar]
- Sage, A.P.; White, C.C.; Siouris, G.M. Optimum systems control. IEEE Trans. Syst. Man Cybern.
**1979**, 9, 102–103. [Google Scholar] [CrossRef] - Ahlberg, J.H.; Nilson, E.N.; Walsh, J.L. The Theory of Splines and Their Applications: Mathematics in Science and Engineering: A Series of Monographs and Textbooks; Elsevier: Amsterdam, The Netherlands, 2016; Volume 38. [Google Scholar]
- Zumbach, G.O. On the Short Term Stability of Financial ARCH Price Processes. Available online: https://arxiv.org/abs/2107.06758 (accessed on 16 December 2021).
- Xing, D.Z.; Li, H.F.; Li, J.C.; Long, C. Forecasting price of financial market crash via a new nonlinear potential GARCH model. Phys. A Stat. Mech. Its Appl.
**2021**, 566, 125649. [Google Scholar] [CrossRef] - Prigogine, I.; Stengers, I. Order Out of Chaos: Man’s New Dialogue with Nature. In Bantam New Age Books; Flamingo: New York, NY, USA, 1985; ISBN 978-0-00-654115-8. [Google Scholar]
- Smith, L. Chaos: A Very Short Introduction; Oxford University Press: Oxford, UK, 2007; 180p. [Google Scholar]
- Kautz, R. Chaos: The Science of Predictable Random Motion; Oxford University Press: Oxford, UK, 2011; ISBN 978-0-19-959458-0. [Google Scholar]
- Yusupov, R.M.; 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] [CrossRef]
- Musaev, A.; Grigoriev, D. Numerical Studies of Statistical Management Decisions in Conditions of Stochastic Chaos. Mathematics
**2022**, 10, 226. [Google Scholar] [CrossRef] - Musaev, A.; Makshanov, A.; Grigoriev, D. Forecasting Multivariate Chaotic Processes with Precedent Analysis. Computation
**2021**, 9, 110. [Google Scholar] [CrossRef] - Musaev, A.; Grigoriev, D. Analyzing, Modeling, and Utilizing Observation Series Correlation in Capital Markets. Computation
**2021**, 9, 88. [Google Scholar] [CrossRef] - Axelsson, O. A generalized conjugate gradient, least square method. Numer. Math.
**1987**, 51, 209–227. [Google Scholar] [CrossRef] - Bolch, B.W.; Huang, C.J. Multivariate Statistical Methods for Business and Economics; Prentice-Hall: Hoboken, NJ, USA, 1974; 317p. [Google Scholar]
- Kim, P.S. Selective finite memory structure filtering using the chi-square test statistic for temporarily uncertain systems. Appl. Sci.
**2019**, 9, 4257. [Google Scholar] [CrossRef] [Green Version] - Fogel, L.J.; Owens, A.J.; Walsh, M.J. Artificial Intelligence through Simulated Evolution, 1st ed.; John Wiley & Sons: New York, NY, USA, 1966; 231p. [Google Scholar]
- Song, W.; Choi, L.C.; Park, S.C.; Ding, X.F. Fuzzy evolutionary optimization modeling and its applications to unsupervised categorization and extractive summarization. Expert Syst. Appl.
**2011**, 38, 9112–9121. [Google Scholar] [CrossRef] - Al-Ahmad, B.; Al-Zoubi, A.M.; Abu Khurma, R.; Aljarah, I. An Evolutionary Fake News Detection Method for COVID-19 Pandemic Information. Symmetry
**2021**, 13, 1091. [Google Scholar] [CrossRef] - Maier, H.R.; Razavi, S.; Kapelan, Z.; Matott, L.S.; Kasprzyk, J.; Tolson, B.A. Introductory overview: Optimization using evolutionary algorithms and other metaheuristics. Environ. Model. Softw.
**2019**, 114, 195–213. [Google Scholar] [CrossRef] - Fernandez, A.; Herrera, F.; Cordon, O.; del Jesus, M.J.; Marcelloni, F. Evolutionary fuzzy systems for explainable artificial intelligence: Why, when, what for, and where to? IEEE Comput. Intell. Mag.
**2019**, 14, 69–81. [Google Scholar] [CrossRef] - Mukhopadhyay, A.; Maulik, U.; Bandyopadhyay, S.; Coello, C.A.C. A Survey of Multiobjective Evolutionary Algorithms for Data Mining. IEEE Trans. Evol. Comput.
**2014**, 18, 4–35. [Google Scholar] [CrossRef] - Huber, P.J. Robust Statistics. In Wiley Series in Probability and Statistics; John Wiley & Sons: New York, NY, USA, 1981; ISBN 978-0-471-41805-4. [Google Scholar]
- Holopainen, M.; Sarlin, P. Toward robust early-warning models: A horse race, ensembles and model uncertainty. Quant. Financ.
**2017**, 17, 1933–1963. [Google Scholar] [CrossRef]

**Figure 1.**Two examples of observing the quotation of the EURUSD currency pair at two 10-day intervals with the system component identified by means of exponential filtering.

**Figure 2.**The process of quote change in the EURUSD currency pair and its five most correlated currency instruments’ quotes.

**Figure 4.**Tonal representation of the EURUSD correlation matrix and its five financial instruments with the highest correlations.

**Figure 7.**Changes in the quotation of a currency instrument and a decision statistic (an oscillator) on a five-day interval.

**Figure 8.**The change in the estimated pairwise correlation between two currency instruments on sliding observation windows of size L = 10, 25, 50, and 75 counts.

**Figure 9.**System component selection using an exponential filter with coefficients α = 0.1 (

**a**) and α = 0.01 (

**b**).

**Figure 10.**Changes in the USDCHF quotation and the group of its five most correlated currency pairs.

**Figure 11.**An example implementation of the simplest control strategy with an adaptive MR oscillator.

**Figure 12.**(

**a**) An example of a suboptimal management strategy using a non-adaptive MR oscillator. (

**b**) The dependence of the gain increase on the generation number for this strategy.

**Figure 13.**(

**a**) An example of a suboptimal control strategy using an adaptive MR oscillator. (

**b**) The dependence of the gain growth on the generation number for this strategy.

Financial Instruments | |||||
---|---|---|---|---|---|

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

EURUSD | EURJPY | EURGBP | EURCHF | EURCAD | USDCAD |

7 | 8 | 9 | 10 | 11 | 12 |

USDCHF | USDJPY | GBPCHF | GBPJPY | GBPUSD | GBPUSD |

13 | 14 | 15 | 16 | 17 | 18 |

AUDUSD | CHFJPY | NZDUSD | NZDJPY | FTSE | DJ |

**Table 2.**Lists of regressors in descending order of correlation on 10-h non-overlapping observation intervals.

Intervals | Regressors | ||||
---|---|---|---|---|---|

1–7 | 1 | 9 | 8 | 10 | 16 |

8–10 | 9 | 1 | 10 | 8 | 16 |

11 | 9 | 10 | 1 | 8 | 16 |

12–14 | 9 | 8 | 10 | 16 | 1 |

15–17 | 9 | 8 | 10 | 16 | 1 |

18–19 | 8 | 9 | 10 | 16 | 4 |

20–21 | 8 | 9 | 10 | 4 | 16 |

22–24 | 9 | 8 | 10 | 4 | 16 |

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**MDPI and ACS Style**

Musaev, A.; Makshanov, A.; Grigoriev, D.
Statistical Analysis of Current Financial Instrument Quotes in the Conditions of Market Chaos. *Mathematics* **2022**, *10*, 587.
https://doi.org/10.3390/math10040587

**AMA Style**

Musaev A, Makshanov A, Grigoriev D.
Statistical Analysis of Current Financial Instrument Quotes in the Conditions of Market Chaos. *Mathematics*. 2022; 10(4):587.
https://doi.org/10.3390/math10040587

**Chicago/Turabian Style**

Musaev, Alexander, Andrey Makshanov, and Dmitry Grigoriev.
2022. "Statistical Analysis of Current Financial Instrument Quotes in the Conditions of Market Chaos" *Mathematics* 10, no. 4: 587.
https://doi.org/10.3390/math10040587