**7. Results**

Empirical properties of three distinct markets will be discussed in this section. We omit the EUR/JPY, and EUR/GBP exchange rates in some of our experiments due to the fact that their properties are greatly similar to the properties of the most traded FX rate EUR/USD. In this case, EUR/USD is selected as the representative exchange rate of the entire FX domain.

### *7.1. Number of Directional Changes*

Equations (16) and (17) connect the expected number of directional changes with parameters of the underlying Brownian motion process. The evolution of real historical returns have properties similar to the Brownian motion. The evolution sometimes compared to the sequence of the free particle moves (see Section 6). Thus, similar counters shown in Figure 2 should be found in heatmaps depicting the number of directional changes empirically registered in real data conditional that the assumption of the normal distribution of real returns is true. EUR/USD, BTC/USD, and SPX500 exchange rates were taken to verify the statement by replicating the same experiment done with the Brownian motion before (Figure 3). A collection of 40 directional-change thresholds ranging from 0.1% to 4.1% defines the scale of the heatmap grid. Colour schemes, used for the plots, have different scales due to the significantly bigger number of directional changes per a period of time in the BTC/USD case. Yellow solid lines indicate the examples of the areas where the number of direction changes is constant. The selected for the examples deltas are *<sup>δ</sup>up* = *δdown* = {1.15%, 2.8%} (EUR/USD, Figure 3a), *<sup>δ</sup>up* = *δdown* = {1.4%, 3.0%} (BTC/USD, Figure 3b), and *<sup>δ</sup>up* = *δdown* = {1.1%, 2.9%} (SPX500, Figure 3c).

**Figure 3.** Heat map of the number of directional changes calculated in (**a**) EUR/USD, (**b**) BTC/USD, and (**c**) SPX500 time series. Each point on the grid represents the number of directional changes registered by a unique pair of thresholds {*<sup>δ</sup>up*, *<sup>δ</sup>down*}. Heatmaps have different scales. Yellow solid lines, specific for each heatmap, label the examples of the areas along which the number of intrinsic events is constant. The dashed lines represent the theoretical areas of the equal number of intrinsic events observed in case of the trend-less time series. White dashed lines are parts of circles centred around the left bottom corner of each picture. The lines go through the intersection of the solid yellow lines and the diagonal of each picture.

Curves in Figure 3a have an almost circular shape and are only slightly shifted towards the bigger *δdown* values. This shift is present due to the downward trend experienced by the exchange rate from 2011 to 2016 (from \$1.4 to \$1.1 per EUR). BTC/USD exchange rate was much more unstable considering that the EUR/USD time series exhibited relative stability with no noticeable regime switches apart from the slow constant price depreciation. The price of Bitcoin grew with accelerating pace by more than 20 times in the second half of 2017 and then lost nearly 70% of its value at the beginning of 201812.

<sup>12</sup> It had a minimum at \$230 per Bitcoin, temporary maximum at about \$20,000, and then a drop to \$6000.

These significant trend changes are pronounced in Figure 3b by yellow contours notably deviated from the circular shape. The price roller-coaster caused considerable disparity of the number of registered directional changes and the ones predicted by Equation (17) (relevant to the trend-less case). As result, the solid price curves can be decomposed into two parts of independent ellipses similar to the ones observed for Brownian motion with non-zero "adjustment coefficient" *γ* (Figure 2b,c).

### *7.2. Realised versus Instantaneous Volatility*

In the second experiment, we compared the annualised volatility computed by the traditional method (Equation (20)) and the volatility based on the observed number of directional changes (Equation (19)).

Returns *Rt* are defined as logarithms of the price change between *St* and *St*−<sup>1</sup> measured over equal periods of time. The number of returns *n* depends on the selected time interval Δ*t* and equal to *n* = *T*/Δ*t* where *T* is the length of the entire tick-by-tick sample. Thus, the length of a sample can be computed ex-ante.

The whole set of returns was used to find the standard deviation of the time series. The measure is also known as realised volatility *σtrad*:

$$R\_t = \ln\left(S\_t / S\_{t-1}\right), \quad R\_{\text{avg}} = \frac{\sum\_{t=1}^{\text{ll}} R\_t}{n}, \quad \sigma\_{\text{tra}} = \sqrt{\frac{\sum\_{t=1}^{n} (R\_t - R\_{\text{avg}})^2}{n-1}}.\tag{20}$$

The directional-change method does not define the number of observations ex-ante in contrast to the traditional approach. According to Equation (19), the size of the directional-change threshold *δ* determines only the expected number of measures (or timestamps) in the data sample of the given length. It is also worth saying that the price moves of the highest frequency, tick-by-tick, do not appear over any predefined period. They occur together with the flow of new orders in the market. The flow, initiated by thousands of independent traders' demands, not synchronised with any periodical process. Thus, the time distance between two consecutive ticks can be represented by a fraction of a second as well as by several minutes. The equally spaced timestamps used to calculate returns for the "natural" estimators have a high chance to happen not at the moment of a new price change. The additional decision should be made on whether the historical price located before the timestamp or right after it should be selected to compute the corresponding return. The directional-change intrinsic time, in turn, directly reacts to the changes of the price levels. This flexibility of the intrinsic time makes it possible to use the data of the highest frequency: tick-by-tick prices.

Specifications of tools used to estimate volatility can affect the experiments results (Müller et al. 1997). Four increasing time intervals Δ*tk*, where *k* = {1, 2, 3, <sup>4</sup>}, were selected to define the distance between each pair of consecutive prices *St* and *St*−<sup>1</sup> used for the "natural estimator": Δ*t*1 = 1 min, Δ*t*2 = 10 min, Δ*t*3 = 1 h, and Δ*t*4 = 1 day. The set of thresholds employed to investigate the directional-change approach can also be arbitrarily chosen. However, we selected them with the intent to compare the results of both experiments. For this reason, we used the number of returns in the data sample corresponding to each time interval Δ*tk* as the target for the number of directional changes registered in the same data set. That is, the collection of four thresholds *δk* was selected in such a way that in the given time series the number of directional changes is approximately equal to the number of time intervals *nk* of the length Δ*tk*. We utilised one of the scaling properties described in Glattfelder et al. (2011) to find the precise thresholds size. The scaling property has the name "time of total-move" scaling law (law 10 in the article). The total-move is composed as the sum of the directional-change (DC) and overshoot (OS) parts. The law connects the size of the threshold *δ* with the waiting time *TTM*(*δ*) between two consecutive intrinsic events:

$$T\_{TM}(\delta) = \left(\frac{\delta}{\mathbb{C}\_{t,TM}}\right)^{E\_{t,TM}},\tag{21}$$

where *Ct*,*TM* and *Et*,*TM* are the scaling coefficients. Equation (21) can be used to express the threshold *δ* in terms of the waiting time *TTM*:

$$
\delta(T\_{TM}) = T\_{TM}^{1/E\_{t,TM}} \mathbb{C}\_{t,TM}.\tag{22}
$$

The currency average scaling parameters *Et*,*TM* and *Ct*,*TM* computed in Glattfelder et al. (2011) are 2.02 and 1.65 × <sup>10</sup>−3, correspondingly. Putting these coefficients into Equation (21), one can calculate that thresholds reciprocal to the selected time intervals Δ*t*1, ... , Δ*t*4 are: *<sup>δ</sup>*(<sup>Δ</sup>*t*1) = 0.013%, *<sup>δ</sup>*(<sup>Δ</sup>*t*2) = 0.039%, *<sup>δ</sup>*(<sup>Δ</sup>*t*3) = 0.095%, and *<sup>δ</sup>*(<sup>Δ</sup>*t*4) = 0.458%. It is worth mentioning that applied scaling parameters are relevant only to the FX market which was the object of the research in Glattfelder et al. (2011). To the extent of our knowledge, parameters specific to Bitcoin prices, as well as to the S&P500 index, were not mentioned in the scientific literature before. Therefore, as the first step, we obtained the parameters by studying the "time of total-move" scaling law of historical Bitcoin, and SPX500 returns. The log-log plot of waiting times *TTM*(*δ*) versus the directional-change threshold size *δ* is provided in Figure 4. The red line marks BTC/USD scaling law and is shown together with black, yellow, and green lines computed for EUR/USD, SPX500, and Geometrical Brownian Motion (GBM) correspondingly. Settings of the latter are chosen to mimic returns typical for the FX market.

**Figure 4.** Time of total-move scaling laws computed for BTC/USD, EUR/USD, SPX500, and Geometrical Brownian Motion (GBM). GBM's parameters are *S*0 = 1.3367, *μ* = 0, *σ* = 20%, *T* = 1 year, and 10 million ticks in total. Scaling parameters *C* and *E* correspond to the coefficients of Equation (21).

Total-move scaling law parameters, obtained in the experiment, exhibit distinct resemblance of the stylised properties of the traditional FX and SPX500, as well as the emerging Bitcoin markets. Scaling factors *Et*,*TM* of EUR/USD, BTC/USD, and SPX500 are 1.827, 1.818, and 1.604, correspondingly. The coefficient specific for the BTC/USD pair is approximately 0.5% smaller than the one of EUR/USD. The coefficient of the SPX500 index is, in turn, is substantially smaller: by 9.9%. The same scaling factor of the GBM is the biggest among others: 1.920 ( ≈ 5.6% difference with EUR/USD). The parameter is noticeably distant from the parameters of the analysed exchange rates. We account the divergence to the non-normal distribution of real returns at ultra-short timescales (fat tails). The fat tails effect is pronounced in Figure 4 as the upward bend of the curves towards the beginning of the X-axis. The bends are read as the longer time needed for a total-move to unwrap than it is predicted by the linear part of the plot in the range of higher thresholds values. Linear regressions, built in the range of straight parts of the curves, are characterised by the scaling coefficients *Et*,*TM*, which are close to the ones observed in GBM. The observed evidence is an additional confirmation of the "Aggregational Gaussianity" stylised fact<sup>13</sup> typical for high-frequency markets (Cont 2001a). Scaling parameters *Ct*,*TM* of EUR/USD, BTC/USD, and SPX500 are 9.07 × <sup>10</sup>−4, 9.94 × <sup>10</sup>−3,

<sup>13</sup> The evidence that the distribution of returns approaches the normal one measured over longer timescales.

and 4.60 × <sup>10</sup>−4, correspondingly. These values are significantly different due to the unlike scale of the corresponding volatility. This volatility dependent scaling parameter is not critical for the current analysis and will be discussed in the future research works.

The goal of the experiment is to compare the volatility computed using the "traditional" approach to the volatility based on the directional-change intrinsic time concept. Scaling law parameters *Et*,*TM* and *Ct*,*TM* of historical BTC/USD returns were used to find the size of the directional-change thresholds, which would result in the average number of registered intrinsic events in the entire data-sample equal to the number of evenly spaced periods *nk*. Expressing the parameter *δk* from the Equation (21) we find that for BTC/USD the thresholds are: *<sup>δ</sup>*(<sup>Δ</sup>*t*1) = 0.09%, *<sup>δ</sup>*(<sup>Δ</sup>*t*2) = 0.33%, *<sup>δ</sup>*(<sup>Δ</sup>*t*3) = 0.89%, *<sup>δ</sup>*(<sup>Δ</sup>*t*4) = 5.13%. The values are about ten times bigger than the ones related to the FX market (mentioned above) because of the proportionally larger realised volatility.

The same procedure, described in the previous paragraph, was performed in order to find the corresponding thresholds for the SPX500 time series. The obtained values: *<sup>δ</sup>*(<sup>Δ</sup>*t*1) = 0.006%, *<sup>δ</sup>*(<sup>Δ</sup>*t*2) = 0.025%, *<sup>δ</sup>*(<sup>Δ</sup>*t*3) = 0.076%, *<sup>δ</sup>*(<sup>Δ</sup>*t*4) = 0.55%.

The set of selected time intervals <sup>Δ</sup>*tk*={1,...,4} and the complementary thresholds *<sup>δ</sup>*Δ*t*1 , ... , *<sup>δ</sup>*Δ*t*4 specific for each considered market were used to calculate realised and instantaneous volatility by traditional and the novel approach. We present in Table 2: average value of the realised volatility *<sup>σ</sup>trad* computed as the sum of all four measurements (*k* = {1, 2, 3, 4}) divided by the number of experiments; its standard deviation *<sup>σ</sup>*<sup>−</sup>*trad*; average value of the instantaneous volatility computed by the novel approach *<sup>σ</sup>DC*; the corresponding standard deviation *<sup>σ</sup>*<sup>−</sup>*DC*; ratios of both measures *<sup>σ</sup>trad*/*<sup>σ</sup>DC* and *<sup>σ</sup>*<sup>−</sup>*trad*/*σ*<sup>−</sup>*DC*. The last column of the table demonstrates the difference in the stability of results obtained by two measures.

The size difference of the realised and the instantaneous volatility is significant and is pronounced across all tested exchange rates (column *<sup>σ</sup>trad*/*<sup>σ</sup>DC*). The realised volatility computed in the "natural" way persistently exceeds the instantaneous volatility discovered via the novel approach. Only the two types of Bitcoin's volatility appear to be 5% different whenever the divergence grows up to 99% in the case of SPX500. The striking difference is partially explained by the various discreteness of the employed data (which will be elaborated in the next section), and partially by the phenomenological properties of the selected markets (more on it in Section 7.4). This phenomenon is captivating especially taking into account that Bitcoin is particularly famous due to its oversized price activity. Its activity is clearly pronounced as the large standard deviation of the instantaneous volatility of BTC/USD pair (column *<sup>σ</sup>*<sup>−</sup>*DC*). Three FX exchange rates, having noticeably smaller realised volatility, are characterised by the wider range of the standard deviation values (column *<sup>σ</sup>*<sup>−</sup>*trad*). The ratio *<sup>σ</sup>*<sup>−</sup>*trad*/*σ*<sup>−</sup>*DC* reaches the 0.02 level computed for EUR/USD. In other words, the standard deviation of the EUR/USD instantaneous volatility is 50 times bigger than the realised volatility value.

**Table 2.** Volatility of the considered time series computed using the "traditional" (Equation (20)) and the directional-change (Equation (19)) approaches. Provided values *<sup>σ</sup>trad* and *<sup>σ</sup>DC* are the average of four measurements performed with specific parameters: in the "traditional" case time intervals between observations *Sn* and *Sn*−<sup>1</sup> are Δ*t*1 = 1 min, Δ*t*2 = 10 min, Δ*t*3 = 1 h, and Δ*t*4 = 1 day. In the case of the directional-change intrinsic time approach, the thresholds *δ* are *<sup>δ</sup>*(<sup>Δ</sup>*t*1) = 0.013%, *<sup>δ</sup>*(<sup>Δ</sup>*t*2) = 0.039%, *<sup>δ</sup>*(<sup>Δ</sup>*t*3) = 0.095%, *<sup>δ</sup>*(<sup>Δ</sup>*t*4) = 0.458% (FX prices), *<sup>δ</sup>*(<sup>Δ</sup>*t*1) = 0.09%, *<sup>δ</sup>*(<sup>Δ</sup>*t*2) = 0.33%, *<sup>δ</sup>*(<sup>Δ</sup>*t*3) = 0.89%, *<sup>δ</sup>*(<sup>Δ</sup>*t*4) = 5.13% (BTC prices), and *<sup>δ</sup>*(<sup>Δ</sup>*t*1) = 0.006%, *<sup>δ</sup>*(<sup>Δ</sup>*t*2) = 0.025%, *<sup>δ</sup>*(<sup>Δ</sup>*t*3) = 0.075%, *<sup>δ</sup>*(<sup>Δ</sup>*t*4) = 0.545% (SPX500).

