*7.4. Volatility Seasonality*

Dacorogna et al. (1993) presented a weekly seasonality pattern of price activity in the FX market. The authors' analysis is based on the assumption that worldwide trading happens at strictly separated time zones with several dominated cores and operates within specific trading hours. Such a physical distribution of traders is embodied in geographical components of the market activity and eventually becomes pronounced as the weekly volatility seasonality. We do not build a similar assumption in our work. Instead, the collection of observed historical returns is treated as the only source of information available for the analysis. Further, we discover and describe the seasonality pattern of instantaneous volatility typical for FX exchange rates, Bitcoin prices, and S&P500 index.

We divide a whole week into a set of 10-min time intervals (bins). There are 1008 equally spaced points located at the fixed distance from the beginning of each week. This is a significantly larger number than the one used in the work Dacorogna et al. (1993) (168 points spaced by one hour intervals). We can afford this decreased granularity thanks to the more detailed historical time series employed for the experiment: instead of 12 million ticks for 26 exchange rates, we have on average 100 million ticks for each of the FX pairs. For each bin, the average number of directional changes will be computed.

The following series of steps allowed to construct the seasonality pattern. First, we run all historical tick-by-tick prices through the directional-change algorithm with the specified threshold *δ*. As soon as a new intrinsic event becomes registered, we check within which out of 1008 bins it happened. We add +1 to the number of directional changes corresponding to that time interval. When there are no prices left in the historical time series, we find the average number of intrinsic events per each bin. Equation (19) is then applied to compute the corresponding instantaneous volatility. Considering the five-year-long historical data, the obtained average is based on nearly 250 observations. Calculated instantaneous volatility values should be later normalised by the number of years and the length of a bin to ge<sup>t</sup> the annualised volatility specific for each bin of the week.

We select the threshold *δ* = 0.01% for the first experiment with FX exchange rates. The average number of directional changes in a week registered by a threshold of this size is approximately equal to the number of 10-min long bins in it (1008). The reconstructed instantaneous volatility seasonality pattern of the FX pairs is shown in Figure 6. The pattern is notably stable across all tested exchange rates and is similar to the one demonstrated in Dacorogna et al. (1993).

**Figure 6.** Instantaneous volatility seasonality of three Forex (FX) exchange rates computed using the directional-change approach (Equation (19)). Applied directional-change threshold *δ* = 0.01%. The whole week is divided by equally spaced time intervals *T* = 10 min (1008 bins in total).

We provide results of the same experiment where the "traditional" volatility estimator (Equation (20)) was employed to reveal the seasonality patterns of the FX exchange rates in Figure 7 and of BTC/USD in Figure 8. In contrast to the volatility seasonality pattern computed using the directional-change approach (Figure 6), the "traditional" pattern is less affected by the frequency of ticks per period of time specific for each studied time series. The difference between the average realised volatility across a week of the most active pair (EUR/JPY) and the least active (EUR/GBP)<sup>14</sup> is equal to 46%. The same difference of the instantaneous seasonality (Figure 6) is 10% bigger and equal to 56%. The "traditional" estimator of the realised volatility seasonality demonstrates more rapid changes in the consecutive bins values. Local maximums at the beginning and the end of a day are considerably abrupt. The reason for this is that the directional-change intrinsic time captures the part of the volatility of the underlying process free of the noise component by ignoring the overshoot part of each trend move. The exact form and scale of the noise component and its connection to the overshoot section of the directional-change intrinsic time is a topic for future research work.

Assets traded in the crypto market have several specific properties which make them noticeably different compared to the traditional financial instruments such as FX exchange rates. Among the characteristics are: open trading within weekends and holidays; the absence of isolated physical trading centres where working hours are fixed; still low acceptance of the emerging market among professional traders. The outlined differences are endorsed by the history of technologies employed in the traditional FX and the emerging crypto worlds. The first one has originated in times when the trading happened in person and the settlement assumed the actual physical assets delivery. The trading organically evolved over time and became digital thanks to the internet expansion. Nevertheless, old properties, such as the governmental and the middle-man controls, have never been removed from the list of the accompanying FX markets design features. Bitcoin, in turn, has been designed as the alternative of the traditional financial system. It benefits from the blockchain technology by endorsing the principles of equality, openness, and accountability. We studied the historical prices of Bitcoin to investigate whether these specialities have any considerable impact on the BTC/USD instantaneous volatility seasonality pattern.

<sup>14</sup> According to the Table 2.

**Figure 7.** Realised volatility seasonality patterns of three FX exchange rates computed using the traditional approach (Equation (20)). Time intervals of 1-min have been used to calculate returns. The size of each bin is 10 min, 1008 bins in total.

**Figure 8.** Realised volatility seasonality patterns of BTC/USD and EUR/USD exchange rates computed using the traditional approach (Equation (20)). Time intervals of 1-min have been used to calculate returns. The size of each bin is 10 min, there are 1008 bins in total.

We apply the same threshold size *δ* = 0.01% used in the FX experiment to compare the seasonality patterns of Bitcoin and EUR. The obtained seasonality pattern put on top of the EUR/USD seasonality is presented in Figure 9. As can be seen from Figure 9, the periodical shape of Bitcoin's curve is much less pronounced in contrast to EUR/USD. Its standard deviation computed within a week is 0.5%. It is roughly four times smaller than the standard deviation of the EUR/USD pattern (equal to 1.9%). Surprisingly, the intra-day maximums and minimums of Bitcoin seasonality do not precisely coincide with those observed in the traditional market. They are shifted towards the time intervals where European and American markets contribute the most to the geographical pattern (as disclosed in Dacorogna et al. (1993)). This observation confirms the one provided in Eross et al. (2017). That is particularly interesting since Asian markets are known for their substantial contribution to the cryptocurrency trading volumes. The fact that China has ruled that financial institutions cannot handle any Bitcoin transactions could be the reason of the observed phenomenon (Ponsford 2015). Instantaneous volatility over weekends is slightly lower than within the middle part of the week and is practically equal to Monday's activity. We attribute the observed facts to the mentioned above non-traditional characteristic of the cryptocurrency market.

**Figure 9.** Instantaneous volatility seasonality of BTC/USD compared to the seasonality pattern of EUR/USD computed using the directional-change approach (Equation (19)). The dark-red curve approximates the Bitcoin seasonality pattern using the Savitzky–Golay filter (number of points in the window is 101, the order of the polynomial is 2). The directional-change threshold *δ* = 0.01% was used in both experiments. Each discrete time interval (bin) is *T* = 10 min. There are 1008 bins in total.

As it was shown before, the instantaneous volatility computed by the novel approach directly depends on the size of the selected directional-change threshold *δ* (Figure 5). To examine the threshold size impact on the seasonality pattern of the real data, we arbitrarily selected the following set of values: *δ* = {0.01%, 0.04%, 0.10%}. The same algorithm described above was applied to reconstruct the volatility seasonality pattern for the FX pair EUR/USD (Figure 10) and SPX500 (Figure 11). The seasonality patterns shift toward higher volatility values when the size of the threshold is bigger. The observation is in line with the results of the experiments on GBM (Figure 5). Average values of EUR/USD seasonality curves computed with thresholds *δ* equal to 0.10% and 0.04% are correspondingly 1.71 and 1.57 times higher than the values computed with *δ* = 0.01%. The dependence of the seasonality smoothness on the size of the directional-change threshold become vividly pronounced: the seasonality curve constructed with the smallest threshold in the set is much sleeker (less wander) than the rest of the curves. This phenomenon should urge researchers and practitioners to select directional-change thresholds according to their needs very carefully while employing the directional-change technique.

**Figure 10.** Volatility seasonality of EUR/USD computed using the novel approach (Equation (19)) and three different thresholds: *δ* = {0.01%, 0.04%, 0.10%}. The size of a bin is 10 min, there are 1008 bins in a week.

**Figure 11.** Volatility seasonality of SPX500 computed using the novel approach (Equation (19)) and three different thresholds: *δ* = {0.01%, 0.04%, 0.10%}. The size of a bin is 10 min, there are 1008 bins in a week.

According to Table 2, realised volatility of Bitcoin returns computed in the "traditional" way is about nine and six times bigger than the analogous volatility of the FX and SPX500 exchange rates (column *<sup>σ</sup>trad*). Besides, the retrieved sample of historical BTC/USD prices has 1.2 million ticks per year, which is 16.7 times smaller than the number of ticks per year in the EUR/USD case (about 20 million). As a result, the choice of the directional-change threshold *δ* has a much more significant effect on the average BTC/USD instantaneous volatility. We demonstrate results of four experiments where different threshold sizes were employed to reveal the seasonality patterns in Figure 12. The same *δ* = 0.01% was used as the reference for the set of all arbitrary selected thresholds: *δ* = {0.01%, 0.03%, 0.10%, 0.20%}. As it can be seen from Figure 12, the increase in the size of *δ* causes the corresponding increase in the volatility level around which the seasonality patterns oscillate. The levels of the seasonality distribution for *δ* = {0.03%, 0.10%, 0.20%} are 1.5, 4.0, and 11.1 times bigger than the value corresponding to the smallest threshold *δ* = 0.01%. The biggest *δ* = 0.20% lifts the value up to the level of *σannual* = 68.5% (which is still smaller than the realised volatility presented in Table 2 (*<sup>σ</sup>annual* = 84.76%)).

**Figure 12.** Instantaneous volatility seasonality of BTC/USD exchange rate computed using the directional-change approach (Equation (19)) and four different thresholds. Applied thresholds, from top to bottom: *δ* = {0.20%, 0.10%, 0.03%, 0.01%}. The dark solid curves approximate the Bitcoin seasonality patterns using the Savitzky–Golay filter (number of points in the window is 101, order of the polynomial is 2). Bin size *T* = 10 min was chosen in all cases (1008 bins in a week). Dashed lines and the numbers over them represent the average level of each seasonality pattern across a week.

More information on the daily instantaneous and realised volatility seasonality ratio is provided in Figures A1–A4 and Table A1 in Appendix A.

### *7.5. Volatility Autocorrelation and Theta Time*

The shape of the persistent instantaneous volatility seasonality patterns computed for the FX and SPX500 exchange rates changes with clear daily periodicity. This observation suggests that there should be a strong autocorrelation of the instantaneous volatility over time. The connection of the number of directional changes and the volatility value (Equation (19)) translates into the autocorrelation of the number of directional changes. We examined the autocorrelation function (ACF) of the number of directional changes observed within each bin of a week to check the assumption. The results of the experiment made for the FX exchange rates are provided in Figure 13. The same size of the directional-change threshold used to reveal the seasonality distribution *δ* = 0.01% was employed. A remarkably stable pattern was found where daily and weekly seasonality is easily recognisable. The ACF function of FX exchange rates discovered in our work is highly similar to the results provided by Dacorogna et al. (1993). Nevertheless, there are clear differences between the FX autocorrelation patterns in our work and in the work of Dacorogna et al. (1993). The ACF of the number of directional changes computed through time lags defined in physical time does not cross the zero level for a much more extended period. It is consistently positive with lags even greater than several weeks. The curve representing the ACF of EUR/JPY has the smallest amplitude (smallest variability). In contrast, curves of EUR/USD together with EUR/GBP invariably follow the same pattern shifted up in the case of EUR/USD.

**Figure 13.** Autocorrelation function of the number of directional changes per 10-min long bins computed in physical time. Vertical dashed lines label weekly intervals. Applied threshold *δ* = 0.01%.

The SPX500 time series ACF has a similar shape to the shape of the FX market ACF. Two distinct properties can be noticed: the higher ACF amplitude and the faster decay. The exact level of decline of all exchange rates will be discussed a few paragraphs later.

The BTC/USD exchange rate seasonality pattern, characterised by much less pronounced instantaneous volatility, has also been tested in order to ge<sup>t</sup> the shape of the autocorrelation function. The results are presented in Figure 13. The amplitude of the ACF curves is the main difference in the values computed for the traditional FX and the emerging BTC/USD markets: the variability of the BTC/USD curve is 10 times lower than the variability of the EUR/USD one. We note that significantly bigger thresholds than the one used in the experiment (*δ* = 0.01%) have also been tested. All results confirmed that they reveal less accurate patterns due to the data insufficiently frequent for the statistical analysis.

A certain level of decline characterises the ACFs of all exchange rates as it can be seen from Figure 13. Large seasonal peaks of the autocorrelation functions drawn against of physical time do not allow to measure the level of decline precisely. A measure capable of converting the stochastic price evolution process to the stationary one should be applied to estimate the level of the downturn better. We minimise the seasonality pattern by employing the concept of theta time ( Θ-time) proposed by Dacorogna et al. (1993). Θ-time is designed to eliminate the periodicity pattern by defining a set of non-equal time intervals within which the measure should be performed. The length of each Θ interval in physical time depends on the historical activity of the market. The theta time concept states that the average cumulative price activity (or volatility) between each consecutive couple of Θ steps is constant. Therefore, the distance between Θ timestamps, measured in physical time, is dictated by the shape of the volatility seasonality pattern. The periods of high price curve activity are equivalent to increasing the speed of physical time. The frequency of Θ stamps increases when the volatility rises too. In contrast, periods of low activity are identical to stretching the flow of the physical time, and the lower number of Θ intervals appears. As a result, active parts of the seasonality pattern, coinciding with the middle of the trading day, have the higher density of Θ timestamps per a unite of the physical time than the standstill sections overnights. Mathematically:

$$
\Theta(t) = \int\_{t\_0}^{t} \sigma(t')dt',\tag{23}
$$

where *t*0 and *t* are the beginning and the end of the considered period of physical time and *σ*(*<sup>t</sup>* ) is the value of the instantaneous volatility corresponding to each moment of the interval. Equation (23) can be transformed into the sum of elements *<sup>σ</sup>*Δ*t* between the beginning and the end of the observed interval Δ*tn* in the case of a non-continuous seasonality pattern where the values are discretely defined in periods Δ*t* (as in our experiment):

$$
\Theta(t) = \sum\_{\Delta t\_0}^{\Delta t} \sigma\_{\Delta t\_n}.\tag{24}
$$

It should be noted that the number of bins in a week is always constant in both physical and Θ times. This is achieved through the assumption that the integral (or the sum) of the weekly activity is the constant value.

The autocorrelation function of the number of directional changes computed in Θ-time is shown in Figure 14. Curves are approximated by the logarithmic function *y* = *AACF* log *x* + *BACF*. The logarithmic coefficients *AACF* and *BACF* are presented on Figure 14 and in Table 3.

Major weekly fluctuations of the volatility seasonality pattern have been successfully eliminated for all three FX and BTC/USD exchange rates. Nevertheless, Θ-time does not completely remove the seasonality shape of ACF in the same way it happened in the work Dacorogna et al. (1993): noticeable peaks are still present in the final part of each business day. Moreover, the SPX500 curve is characterised by vividly pronounced daily seasonality pattern despite being run through the theta time algorithm. The phenomenon, observed in the original paper (Dacorogna et al. 1993), was explained by the non-optimal setup of the chosen model. The assumed same activity for all working days is indeed not fully correct (see Figure 6). However, we do not use any analytical expression postulating equal daily activity to describe the seasonality pattern. Instead, components *<sup>σ</sup>*Δ*t* of real empirically found volatility seasonality patterns depicted in Figures 6 and 9 were utilised to define the timestamps in Θ-time. Therefore, we eliminate the inefficiency connected to the assumption mentioned above. Thus, the alternative interpretation for the remained seasonality should be provided.

**Figure 14.** Autocorrelation function (ACF) of the number of directional changes per a bin in Θ-time. Vertical dashed line labels one week interval. There are 1008 bins in a week.

**Table 3.** Parameters of the logarithmic decay *y* = *AACF* log *x* + *BACF* used to fit the autocorrelation function (ACF) of the number of directional changes in Θ-time (Figure 14).


We attribute the remaining fluctuations to the selected directional-change algorithm, which dissects the price curve into a collection of alternating trends. We also claim that the choice of the frequency of bins in a week used for the experiments affects the shape of the autocorrelation function in theta time. According to the directional-change algorithm (see Section 1), the dissection procedure has to be initialised only once and then it performs unsupervised. The evolution of the price curve dictates the sequence of intrinsic events. This fact leads to a certain dilemma: once registered, to which bin of a week should the intrinsic event be assigned? The following example illustrates the preditacament. A couple of prices, at which two subsequent directional changes become registered, could belong to different bins. Let us say these are the intervals Δ*tn*−<sup>1</sup> and Δ*tn*. This means that the beginning of the price move that triggered the latest intrinsic event had started within Δ*tn*−1. But the end of this price trajectory finishes within the interval Δ*tn*. The crucial point is at what part of the Δ*t* are the beginning and the end located. In the extreme case, the whole price trajectory before the directional change could be fully placed inside of the interval Δ*tn*−1. The latest tick that eventually triggered the new directional-change event can be at the very beginning of Δ*tn*. Should such an event be assigned to the bin Δ*tn*−<sup>1</sup> or to Δ*tn*? The answer to this question is particularly important considering the effect the threshold size has on the seasonality patterns (Figures 10 and 11). The patterns constructed by using different thresholds have not only different average value over a week but also characterised by slightly shifted regions of local maximums and minimums (see, for example, the curves for *δ* = 0.01% and *δ* = 0.10%).

A better way of associating locations of intrinsic events with bins of a week is another question related to the transition from the physical to intrinsic time and vice versa. This topic should be discussed in more details in further research works. Until then, the use of smaller thresholds and bigger time intervals is the strategy capable to impact the localisation problem positively.
