**8. Concluding Remarks**

The language of traditionally considered drawdowns and drawups has been translated into the language of the directional-change intrinsic time. This translation made it possible to interpret the evolution of a price curve as a sequence of alternating trends of the given scale. The observed number of directional changes per period of time has been connected to the properties of the studied time series characterised by the instantaneous volatility *σDC* and the trend *μ*. The choice of directional-change thresholds *<sup>δ</sup>up* and *δdown* used to dissect the historical price curve is arbitrary but affects the results of the experiment. Bigger thresholds tend to register higher instantaneous volatility than the smaller ones. Equations (10), (11) and (16), connecting the observed number of directional changes to the properties of the studied process, have been validated by a Monte Carlo simulation. The simulation confirmed the robustness and accuracy of the obtained analytical expressions.

We extended the work of Dacorogna et al. (1993) by discovering the instantaneous volatility weekly seasonality pattern. One representative of the emerging cryptocurrency markets, Bitcoin, as well as of the traditional and widely accepted FX (EUR/USD, EUR/GBP, EUR/JPY) and stock (S&P500) markets, were considered. The connection of the number of directional-change intrinsic events to the instantaneous volatility has been employed to perform the computation. BTC/USD and SPX500 "total-move scaling laws" were computed for the first time to facilitate the seasonality discovery.

Similar patterns of the realised and the instantaneous volatility were obtained. Several noticeable differences between the results demonstrated in the work Dacorogna et al. (1993) and the ones presented in the current paper have been highlighted. First, the method based on the directional-change intrinsic time concept significantly simplifies the construction of the instantaneous volatility seasonality pattern operating with the information of the highest resolution (tick-by-tick prices). Second, the autocorrelation function of the number of directional changes computed in physical time stays positive for a notably long period of time. Third, the beginning of the volatility autocorrelation function computed in Θ-time can be approximated by the logarithmic function. The part of the autocorrelation plot after the lag bigger than one week declines linearly.

The difference between the instantaneous volatility seasonality patterns of the traditional FX and emerging Bitcoin markets demonstrates the currency digitisation and globalisation effect. The effect can be considered as the template for the future analogous markets to come.

The insights provided within this paper underline the relevance of the proposed directional-change framework as a valuable alternative to the traditional time-series analysis tools. The independence of directional-change intrinsic time on the frequency of price changes over a period of time makes it an effective tool for capturing periods of changing price activity. Results of the provided research can be used to extend the set of risk managemen<sup>t</sup> tools constructed to evaluate the statistical properties of traditional and emerging financial markets.

**Author Contributions:** Conceptualisation and methodology, A.G., V.P. and R.O.; software and validation, V.P.; investigation, V.P. and A.G.; writing—original draft preparation, V.P.; writing—review and editing, V.P. and A.G.; and supervision, A.G. and R.O.

**Funding:** This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie gran<sup>t</sup> agreemen<sup>t</sup> No. 675044.

**Acknowledgments:** We thank James Glattfelder, flov technologies, for all comments that greatly improved the manuscript. We are also grateful to the reviewers for all remarks.

**Conflicts of Interest:** The authors declare no conflict of interest.

**Appendix A. Daily Seasonality**

**Figure A1.** Daily realised volatility ratio of BTC/USD measured over 10-min time intervals dissecting the entire week into 1008 bins. The volatility is computed according to the "traditional" approach (Equation (20)).

**Figure A2.** Daily realised volatility ratio of the given exchange rates (labelled by \*\*\*) and EUR/USD. The volatility is computed according to the "traditional" approach (Equation (20)).

**Figure A3.** Daily instantaneous volatility ratio of BTC/USD measured over 10-min time intervals dissecting the entire week into 1008 bins. The volatility is computed according to the novel approach (Equation (19)).

**Figure A4.** Daily instantaneous volatility ratio of the given exchange rates (labelled by \*\*\*) andEUR/USD.Thevolatilityiscomputedaccording to thenovelapproach(Equation(19)).

**Table A1.** Daily instantaneous volatility ratio and the weekly standard deviation of two FX (EUR/JPY and EUR/GBP), one crypto (BTC/USD), and one stock (SPX500) exchange rates to EUR/USD. Columns *Ratio*∗ stands for the ratio of the average daily volatility of the corresponding exchange rate and one of EUR/USD. The average value computed over 10-min intervals. Columns *std*∗ contain the standard deviation values of the volatility ratio over the set of 10-min intervals. The subscripts *trad* and *DC* label the measures made using the traditional volatility estimator (Equation (20)) and the novel approach (Equation (19)) correspondingly. The daily volatility ratios are graphically presented in Figures A2 and A3.

