**3. Directional-Change Intrinsic Time**

The existing literature on risk-management techniques primarily relies on physical time as a measure of the length and periodicity of financial events. In other words, the existence of a universal clock dictating the evolution of the market prices is assumed. However, the volatilities of different time resolutions behave differently (Müller et al. 1997). The volatility size depends on the scale of the entire time series as well as on the moment when the price activity started to be observed. More robust techniques which are beyond the limits of physical time are needed to handle this stochasticity.

The concept of directional-change intrinsic time (Guillaume et al. 1997) is one of the methods capable of replacing the universal physical clock with intrinsic one. This is an event-based framework which considers the activity of market prices as the indicator of the transition between its different states. The framework dissects a price curve into a collection of sections characterised by alternating trends of the arbitrary defined size. The essence of the concept is closely related to the meaning of drawdowns and drawups: the collection of directional changes following each other can be interpreted as the alternating sequence of drawdowns and drawups. The frequency of price changes in physical time does not play any role in the directional change dissection procedure.

The concept of trend directional changes provided by Guillaume et al. (1997) is capable of connecting the continuous flow of physical time with the endogenous evolution of price returns. According to the event-based space proposed by Guillaume, only a sequence of price trends continuously alternating in direction has to be considered. The price curve gets dissected into a collection of alternating drawups and drawdowns or trend rises and trend falls correspondingly. Each elementary trend ends once a new price curve reversal is observed. Continuous price moves towards the direction of the latest trend change are called overshoots. The current state of the system changes only at the moments when the trend of the given size reverses its direction. Thus, the set of intrinsic events is decoupled from the flow of physical time. Instead, it depends only on the size of considered drawups and drawdowns labelled by the threshold *δ*. An example of a price curve dissected into a collection of directional changes is provided in Figure 1.

**Figure 1.** A part of EUR/USD price curve (grey) dissected into a set of directional-changes (grey squares) using a directional-change threshold *δ*. The size of the arbitrary chosen threshold is presented in the middle of the figure. Grey circles mark local extremes between two consecutive directional changes. The vertical distance between each directional-change and preceding extreme price is bigger or equal to the size of the threshold *δ*. Vertical dashed lines indicate the end of each trend section (identified only after the next event becomes observed) and go through the local extremes (circles). The timeline below the plot contains equal time intervals *T*1, *T*2, *T*3 and length of each directional-change section *<sup>T</sup>*1(*δ*),..., *<sup>T</sup>*6(*δ*).

The density of directional-change intrinsic events depends only on the price curve evolution and the considered trend size. The stochastic nature of price evolution results in the phenomenon depicted in Figure 1: non-equal number of intrinsic events (empty squares) correspond to the equal periods of physical time. The physical interval *T*1 contains only the end of the Section 1 (sections coincide with the intrinsic events and are separated by the dashed vertical lines) while the equal interval *T*2 hosts three segments, namely 2, 3, 4. This property of directional-change intrinsic time can be engaged as the efficient noise filtering technique: the intrinsic time ignores price changes between directional change. At the same time, it allows us to efficiently capture the most relevant to risk managemen<sup>t</sup> information: precise moments of all trend changes. The equally spaced time intervals typically employed in the financial analysis are not capable of doing anything of the above: price timestamps, evenly spaced through periods *T*1, *T*2, and *T*3, do not contain information on the extreme price curve activity located in the period *T*2. This disability of the traditional price analysis techniques over stochasticity of the market's activity develops into volatility estimators that are too stiff and biased.

The concept of directional-change intrinsic time, applied for studying historical price, returns reveals multiple statistical properties of high-frequency markets. Guillaume et al. (1997) were the first researchers to uncover a scaling law<sup>1</sup> relating the expected number of directional-changes *N*(*δ*) observed over the fixed period to the size of the threshold *δ*. Mathematically:

$$N(\delta) = \left(\frac{\delta}{\mathbb{C}\_{N,DC}}\right)^{E\_{N,DC}} \text{ .}\tag{4}$$

where *C N*,*DC* and *EN*,*DC* are the scaling law coefficients. Glattfelder et al. (2011) employed the directional-change framework to discover 12 independent scaling laws which hold across three orders of magnitude and are present in 13 currency exchange rates. Later Golub et al. (2017) described a successful trading strategy exploiting a collection of tools build upon directional-change intrinsic time. The proposed strategy is characterised by the annual Sharpe ratio greater than 3.0. The persistence of revealed scaling laws became the base elements for the tools designed to monitor market's liquidity at multiple scales (Golub et al. 2014).
