# Numerical Studies of Channel Management Strategies for Nonstationary Immersion Environments: EURUSD Case Study

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

_{k}= x

_{k}+ v

_{k}, k = 1,…,n

_{k}, k = 1,…,n is a system component formed by sequentially smoothing the time series of initial observations y

_{k}, k = 1,…,n and used in the process of making management decisions, and v

_{k}, k = 1,…,n is the noise.

_{k+1}= Φ

_{k+1/k}x

_{k}+ w

_{k}, k = 1,…,n, where w

_{k}is the noise process that imitates the admissible errors in the system.

- Their system component x
_{k}, k = 1,…,n is an oscillatory nonperiodic process with a large number of local trends. This description indicates the possibility of interpreting this process as an implementation of the dynamic chaos model [21,22,23,24,25,26,27,28,29]. However, the proof of this statement requires a strict formalization of the discrimination of deterministic chaotic and non-stationary random processes, which will require additional research.

_{k}= x

_{k}± B, k = 1,…,n. Variations of observations inside the channel |y

_{k}− x

_{k}| = |δy

_{k}| ≤ B are interpreted as fluctuations that do not contain a pronounced trend, and the process itself is sometimes called a sideways trend or a flat. The choice of the channel width can be driven by various considerations. It usually lies in the range (1–3)s

_{y}, where s

_{y}is the estimate of the standard deviation (SD) δy

_{k}, k = 1,…,n. In general, the choice of channel width is an option that depends on the features of the TP management. In some cases, it can be a variable value B

_{k}= B

_{k}(y

_{k}), k = 1,…,n.

_{k}, k = 1,…,n breaking out of the channel is interpreted as the emergence of a trend in some management strategies. In the case of “playing by the trend”, this invokes a recommendation to open a position in the direction corresponding to the sign of the channel boundary. The position can be closed when a given level of gain (TP, “take profit”) or loss (SL, “stop loss”) is reached, or in accordance with other, more flexible rules defined by the management strategy.

_{j}, j = 1, …, M corresponding to it that provide maximum profit:

_{open},k

_{close})

_{j}, j = 1,…,M, and, in some cases, the lot size. If the resulting amount at some k-th step turns out to be less than the trader’s available deposit R

_{0}, then this means a complete loss. In order to isolate the system component, any technique of sequential filtration can be applied. In the simplest case, an exponential filter is used for this purpose, defined as [31]:

_{k}= αy

_{k}+ (1 − α)y

_{(k-1)}= x

_{(k-1)}+ α(y

_{k}− x

_{(k-1)}), k = 2,…, n

#### 2.1. Channel Strategy Based on Moving by the Trend

_{k}, k = 1,…,n be the observed monitoring process of an asset quotation with a given duration n. The system component x

_{k}, k = 1,…,n consists of an exponential filter (3) with a given smoothing coefficient α. The channel width is denoted as B, measured in pipses. Similarly, the levels of profit TP and the level of acceptable loss SL are set. The values of α, B, TP and SL are optional parameters. The choice of these parameters depends on the knowledge and intuition of the trader, as they completely determine the effectiveness of management. However, in trading, intuition and other human abilities often turn out to be ineffective. Thus, a need for strictly formalized and mathematically sound solutions arises.

_{k}> x

_{k}+ B or Open Dn at y

_{k}< x

_{k}− B, k = 1,…,n.

_{k-1}≤ (x

_{k-1}+ B) & (y

_{k}> x

_{k}+ B) or y

_{k-1}≥ (x

_{k-1}+ B) & (y

_{k}< x

_{k}+ B), k = 1,…,n. Otherwise, a position will be opened at each step outside the channel. The position is closed either when the y

_{k}= y

_{close}> y

_{open}+ TP or y

_{k}= y

_{close}< y

_{open}− SL levels are reached (with Open Up) or y

_{close}< y

_{open}− TP or y

_{close}> y

_{open}+ SL (with Open Dn).

_{k}− x

_{k}−

_{nW}, k = 1,…,n, nW being the size of the sliding observation window, changes to the opposite. However, this article is focused on the analysis of the fundamental features of channel management strategies, and, as a result, omits various complications.

#### 2.2. Channel Strategy Based on Moving against the Trend

_{k}> x

_{k}− B) & (y

_{k-1}< x

_{k-1}− B), k = 2,…,n, and Open Dn − (y

_{k}< x

_{k}+ B) & (y

_{k-1}≥ x

_{k-1}+ B), k = 2,…,n. The position is closed when either of the levels y

_{close}> y

_{open}+ TP; y

_{close}< y

_{open}− SL are reached with Open Up or y

_{close}< y

_{open}− TP; y

_{close}> y

_{open}+ SL with Open Dn.

_{k}, k = 1,…,n.

## 3. Results

_{k}, k = 1,…,n is formed by exponential filter (3) with α = 0.05. The width of the channel B = ±10 p is measured in points or pipses (p). The level of profit and the level of acceptable loss are TP = SL = 10 p.

#### 3.1. Evaluation of CSF’s Potential Performance

_{Dn}and B

_{Up}, and the stop parameters TP and SL. Let us set the ranges of the selected parameters to α = 0.01:0.01:0.15, B

_{Dn}, B

_{Up}= 10:1:15, TP, SL = 10:1:15. For the selected three-day observation segment considered in the previous example, the best result of CSF was R = 166 p with the parameters P* = (α, B

_{Dn}, B

_{Up}, TP, SL)* = (0.1, 13, 16, 23, 16). The implementation of the management strategy with the specified parameters at the selected observation interval is shown in Figure 9, and the changes in the overall result of management are shown in Figure 10.

_{Dn}, B

_{Up}and the filtration coefficient α. It is possible that this strategy will be more effective with separate dynamic adjustment of the upper and lower boundaries B

_{Dn}, B

_{Up}, regarded as functions of the rate of change of the system component estimated on the sliding observation window B

_{Dn}, B

_{Up}= F(x

_{k}− x

_{k-τ}), k = τ + 1,…,n. This issue requires additional research.

_{k}, k = 1,…,n is evaluated as the result of the smoothing performed by exponential filter (3) with α = 0.05. The initial values of the option parameters are: channel width B = ±10 p, take profit level TP = 10 p, acceptable loss level SL = 10 p.

_{k}, k = 1,…,n. In Figure 1, Figure 8, Figure 11 and Figure 13, it can be easily seen that when a strong trend occurs, the system component formed by the smoothing filter lags behind the observation process, which leads to a shift of the entire channel and incorrect use of the management strategy concept itself. Hence, natural suggestions arise for improving the management by modifying the sequential filtering procedure and dynamically changing the size of the lower and upper channel boundaries.

#### 3.2. Evaluation of CSB’s Potential Performance

_{Dn}, B

_{Up}= 5:1:15, TP, SL = 7:1:15. Using a brute force search, the best result of the CSF strategy was R = 166p with P* = (α, B

_{Dn}, B

_{Up}, TP, SL)* = (0.1, 13, 16, 23, 16). The implementation with the specified parameters at the selected observation interval is shown in Figure 11.

_{Dn}, B

_{Up}, TP, SL)* = (0.03, 8, 5, 17, 17). Its implementation is shown in Figure 15.

- Both strategies have profitable decisions, but the result in both cases is not stable and small changes in control parameters can lead to a radical decrease in gain;
- Management against the trend needs a higher level of smoothing, which activates the opening of a position on a sideways trend and reduces the frequency of management in areas with a strong trend. On the contrary, playing by the trend requires reducing the degree of smoothness, which makes it possible to more effectively detect a strong trend and activate management in these areas of observation;
- The width of the channel when playing against the trend is significantly, two to three times, smaller than in the opposite case, which increases the frequency of opening in horizontal quotation sections. Obviously, in both cases, the choice of channel width is related to the degree of volatility, as a measure of which, for example, the standard deviation (SD) of observations relative to the system component can be used.

#### 3.3. Parametric Stability of Optimal Solutions

_{Dn}, B

_{Up}, TP, SL)* = (0.02, 7, 14, 8, 19).

_{Dn}, B

_{Up}, TP and SL from optimal in the range P* ± 0.1 P*, and α in range α* ± 0.5 α*. The range of changes was divided into eight steps, i.e., four steps in each direction from the optimal value of the parameter. Table 1 presents the changes in performance for this strategy.

#### 3.4. Examining the Dynamic Stability of Optimal Solutions

_{Dn}, B

_{Up}, TP, SL)* = (0.03, 7, 6, 11, 12), the use of which produces gain R = 164 p.

_{Dn}, B

_{Up}= 5:1:15, TP, SL = 7:1:15. The test result of using CSB was R = 475 p with P* = (α*, B

_{Dn}*, B

_{Up}*, TP*, SL*) = (0.02, 6, 8, 17, 21). The performance of the management strategy with specified parameters at the selected observation interval is shown in Figure 20.

_{Dn}, B

_{Up}= 5:1:15, TP, SL = 7:1:15. Their best values, P* = (α*, B

_{Dn}*, B

_{Up}*, TP*, SL*) = (0.1, 11, 14, 16, 13), were obtained by brute force search.

## 4. Discussion

- Dynamic adaptation of decision-making algorithms with the choice of the best parameters on sliding observation windows;
- Robustification, i.e., reduction of the sensitivity of solutions to changes in the dynamic characteristics of the observed processes;
- Improvement of management based on more flexible computational schemes for selecting boundaries of a channel or several channels and decision-making algorithms;
- Structural adaptation, for example, based on switching between management strategies as a reaction to changes in the dynamic properties of the system component of the observation series;
- Self-organization based on artificial intelligence, which independently creates strategies that are not contained in the source code of the management program;
- The use of composite algorithms combining the capabilities of robustification and adaptation in management decision making;
- The use of external add-ons that carry information exogenous to technical analysis on expected trends of the considered financial instrument and market mood in general.

## 5. Conclusions

- CSF and CSB strategies are strict alternatives. Nevertheless, in the absence of prior information about the expected dynamics of the observed process, both strategies, as a rule, lead to losses on the same trading intervals;
- Posterior parametric optimization of both CSF and CSB management strategies has shown that they can be profitable in almost any observation area. This result is quite paradoxical, because these strategies are strict alternatives;
- With a shift of the observation interval equal to one full day, the strategy optimized on the previous interval becomes losing again. This leads to a fairly obvious conclusion about the priority of the problem of management stability in an unstable immersion environment;
- Small time shifts of observation series lead to a loss of optimality of the management algorithm parameters; however, the overall positive result of management is preserved in most cases. This means there is feasibility to using computational schemes with sequential parametric adaptation of the management model;
- Parametric adaptation based on brute forcing the values of optional parameters produces an optimal solution. However, the computational load increases exponentially with the increase in the dimension of the parameter vector. If the time shifts between the observation intervals on which the model is optimized are small, the complexity can go beyond the limits even for high-performance computers. This implies the recommendation to switch to suboptimal computational schemes for adapting the management model;
- As a variant of suboptimal adaptation of the management model, we propose to use algorithms based on evolutionary modeling [36]. These studies have been completed and are being prepared for publication.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ying, Q.; Yousaf, T.; Ain, Q.U.; Akhtar, Y.; Rasheed, M.S. Stock investment and excess returns: A critical review in the light of the efficient market hypothesis. J. Risk Financ. Manag.
**2019**, 12, 97. [Google Scholar] [CrossRef] [Green Version] - Rajablu, M. Value investing: Review of Warren Buffett’s investment philosophy and practice. Res. J. Financ. Account.
**2011**, 2, 1–12. [Google Scholar] - Bartram, S.M.; Grinblatt, M. Agnostic fundamental analysis works. J. Financ. Econ.
**2018**, 128, 125–147. [Google Scholar] [CrossRef] [Green Version] - Nti, I.K.; Adekoya, A.F.; Weyori, B.A. A systematic review of fundamental and technical analysis of stock market predictions. Artif. Intell. Rev.
**2020**, 53, 3007–3057. [Google Scholar] [CrossRef] - Zapranis, A.; Tsinaslanidis, P.E. Identifying and evaluating horizontal support and resistance levels: An empirical study on US stock markets. Appl. Financ. Econ.
**2012**, 22, 1571–1585. [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]
- Niederhoffer, V.; Kenner, L. Practical Speculation; John Wiley & Sons: New York, NY, USA, 2005. [Google Scholar]
- Colby, R.W.; Meyers, T.A. The Encyclopedia of Technical Market Indicators; IRWIN Professional Publishing: Burr Ridge, IL, USA, 2012. [Google Scholar]
- Chordia, T.; Goyal, A.; Saretto, A. p-Hacking: Evidence from Two Million Trading Strategies; No. 17-37; Swiss Finance Institute: Zurich, Switzerland, 2018. [Google Scholar]
- 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]
- Zhang, M.H.; Cheng, Q.S. Gaussian mixture modelling to detect random walks in capital markets. Math. Comput. Model.
**2003**, 38, 503–508. [Google Scholar] [CrossRef] - de Wolff, T.; Cuevas, A.; Tobar, F. Gaussian process imputation of multiple financial series. In Proceedings of the ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Barcelona, Spain, 4–8 May 2020; pp. 8444–8448. [Google Scholar]
- McGroarty, F.; Booth, A.; Gerding, E.; Chinthalapati, V.R. High frequency trading strategies, market fragility and price spikes: An agent based model perspective. Ann. Oper. Res.
**2019**, 282, 217–244. [Google Scholar] [CrossRef] [Green Version] - Tykocinski, O.; Israel, R.; Pittman, T.S. Inaction inertia in the stock market. J. Appl. Soc. Psychol.
**2004**, 34, 1166–1175. [Google Scholar] [CrossRef] - Schmitt, N.; Westerhoff, F. Heterogeneity, spontaneous coordination and extreme events within large-scale and small-scale agent-based financial market models. J. Evol. Econ.
**2017**, 27, 1041–1070. [Google Scholar] [CrossRef] [Green Version] - King, T.; Koutmos, D. Herding and feedback trading in cryptocurrency markets. Ann. Oper. Res.
**2021**, 300, 79–96. [Google Scholar] [CrossRef] - Downey, A.B. Think Bayes: Bayesian Essentials with R, 2nd ed.; Springer: New York, NY, USA, 2014. [Google Scholar]
- Stone, J.V. Bayes’ Rule: A Tutorial Introduction to Bayesian Analysis; Sebtel Press: Sheffield, UK, 2013; p. 174. [Google Scholar]
- Nyberg, S.J. The Bayesian Way: Introductory Statistics for Economists and Engineers; John Wiley & Sons: Hoboken, NJ, USA, 2018. [Google Scholar]
- Kalman, R.E. A New Approach to Linear Filtering and Prediction Problems. J. Basic Eng.
**1960**, 82, 35–45. [Google Scholar] [CrossRef] [Green Version] - Guanrong, C. Chaos theory and applications: A new trend. Chaos Theory Appl.
**2021**, 3, 1–2. [Google Scholar] - Gardini, L.; Grebogi, C.; Lenci, S. Chaos theory and applications: A retrospective on lessons learned and missed or new opportunities. Nonlinear Dyn.
**2020**, 102, 643–644. [Google Scholar] [CrossRef] - Davies, B. Exploring Chaos: Theory and Experiment (Studies in Nonlinearity); CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Feldman, D. Chaos and Dynamical Systems; Princeton University Press: Princeton, NJ, USA, 2019. [Google Scholar]
- Faggini, M.; Parziale, A. The failure of economic theory. Lessons from Chaos Theory. Mod. Econ.
**2012**, 3, 16802. [Google Scholar] [CrossRef] [Green Version] - Glenn, E.J. Chaos Theory: The Essentials for Military Applications; The Newport Papers; CreateSpace Independent Publishing Platform: Scotts Valley, CA, USA, 2012. [Google Scholar]
- Rosenthal, C.; Jones, N. Chaos Engineering. System Resiliency in Practice; O′Reilly Publishing: Newton, MA, USA, 2020. [Google Scholar]
- Skiadas, C.H.; Skiadas, C. Handbook of Applications of Chaos Theory; Chapman and Hall: London, UK; CRC: London, UK, 2016. [Google Scholar]
- Sundbye, L. Discrete Dynamical Systems, Chaos Theory and Fractals; CreateSpace Independent Publishing Platform: Scotts Valley, CA, USA, 2018. [Google Scholar]
- Musaev, A.A.; Makshanov, A.V.; Grigoriev, D.A. Forecasting Multivariate Chaotic Processes with Precedent Analysis. Computation
**2021**, 9, 110. [Google Scholar] [CrossRef] - Gardner, E.S., Jr. Exponential smoothing: The state of the art—Part II. Int. J. Forecast.
**2006**, 22, 637–666. [Google Scholar] [CrossRef] - Musaev, A.A.; Grigoriev, D.A. Analyzing, Modeling and Utilizing Observation Series Correlation in Capital Markets. Computation
**2021**, 9, 88. [Google Scholar] [CrossRef] - Musaev, A.A.; Borovinskaya, E.E. Evolutionary Optimization of Case-Based Forecasting Algorithms in Chaotic Environments. Symmetry
**2021**, 13, 301. [Google Scholar] [CrossRef] - Maronna, R.A.; Martin, R.D.; Yohai, V.J.; Salibián-Barrera, M. Robust Statistics: Theory and Methods (with R); John Wiley & Sons: Hoboken, NJ, USA, 2019. [Google Scholar]
- Musaev, A.; Grigoriev, D. Multi-expert Systems: Fundamental Concepts and Application Examples. J. Theor. Appl. Inf. Technol.
**2022**, 100, 336–348. [Google Scholar] - Pejić Bach, M.; Krstić, Ž.; Seljan, S.; Turulja, L. Text mining for big data analysis in financial sector: A literature review. Sustainability
**2019**, 11, 1277. [Google Scholar] [CrossRef] [Green Version] - Musaev, A.; Grigoriev, D. Numerical Studies of Statistical Management Decisions in Conditions of Stochastic Chaos. Mathematics
**2022**, 10, 226. [Google Scholar] [CrossRef]

**Figure 5.**Three-dimensional plots of the management results for the selected ranges of optional parameters.

**Figure 6.**Network approximation of Figure 5.

**Figure 16.**Example implementation of CSB with five optimized parameters on subsequent three-day interval.

**Figure 17.**Performance of optimized control after applying the CSB strategy in the observation area with optimization (

**a**); and in the subsequent interval of the same duration (

**b**).

**Figure 18.**Example implementation of CSB during a single observation day with optimal optional parameters.

k | α | R | B_{Dn} | R | B_{Up} | R | TP | R | SL | R |
---|---|---|---|---|---|---|---|---|---|---|

1 | 0.0100 | 50 | 6.300 | 244 | 12.60 | 170 | 7.2 | 250 | 17.10 | 229 |

2 | 0.0125 | 80 | 6.475 | 260 | 12.95 | 203 | 7.4 | 250 | 17.57 | 229 |

3 | 0.0150 | 202 | 6.650 | 270 | 13.30 | 241 | 7.6 | 250 | 18.05 | 246 |

4 | 0.0175 | 240 | 6.825 | 263 | 13.65 | 247 | 7.8 | 250 | 18.52 | 246 |

5 | 0.0200 | 267 | 7.000 | 267 | 14.00 | 267 | 8.0 | 267 | 19.00 | 267 |

6 | 0.0225 | 238 | 7.175 | 268 | 14.35 | 258 | 8.2 | 272 | 19.47 | 267 |

7 | 0.0250 | 203 | 7.350 | 261 | 14.70 | 248 | 8.4 | 272 | 19.95 | 267 |

8 | 0.0275 | 148 | 7.525 | 262 | 15.05 | 245 | 8.6 | 272 | 20.42 | 263 |

9 | 0.0300 | 96 | 7.700 | 257 | 15.40 | 238 | 8.8 | 272 | 20.90 | 263 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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.
Numerical Studies of Channel Management Strategies for Nonstationary Immersion Environments: EURUSD Case Study. *Mathematics* **2022**, *10*, 1408.
https://doi.org/10.3390/math10091408

**AMA Style**

Musaev A, Makshanov A, Grigoriev D.
Numerical Studies of Channel Management Strategies for Nonstationary Immersion Environments: EURUSD Case Study. *Mathematics*. 2022; 10(9):1408.
https://doi.org/10.3390/math10091408

**Chicago/Turabian Style**

Musaev, Alexander, Andrey Makshanov, and Dmitry Grigoriev.
2022. "Numerical Studies of Channel Management Strategies for Nonstationary Immersion Environments: EURUSD Case Study" *Mathematics* 10, no. 9: 1408.
https://doi.org/10.3390/math10091408