*6.1. Waiting Time*

We model the set of prices {*St* : *t* ≥ 0} as an arithmetic Brownian motion with trend *μ* and volatility *σ*:

$$dS\_t = \mu dt + \sigma dB\_t.\tag{5}$$

In terms of the directional-change intrinsic time framework, *Tup*(*<sup>δ</sup>up*) denotes the time for an upward directional change of the size *<sup>δ</sup>up* > 0 to unfold. In other words, it is the time interval which passes until the price increases by *<sup>δ</sup>up* percents from the local minimum *mt*. Technically:

$$T\_{up}(\delta\_{up}) = \inf\{t > 0 : \frac{\mathbb{S}\_t - m\_t}{m\_t} \ge \delta\_{up}\},\tag{6}$$

where

$$m\_{\ell} := \inf\_{\mathcal{C} \in [0, t]} S\_{\mathcal{C}}.\tag{7}$$

Similarly, *Tdown*(*<sup>δ</sup>down*) is the time of a downward directional change of the size *δdown* > 0:

$$T\_{down}(\delta\_{down}) = \inf\{t > 0 : \frac{M\_t - S\_t}{M\_t} \ge \delta\_{down}\},\tag{8}$$

where

$$\mathcal{M}\_{\mathbf{f}} := \sup\_{\mathbf{c} \in [0, t\_{\parallel}]} \mathcal{S}\_{\mathbf{c}}.\tag{9}$$

Both of these equations are also known in the literature as waiting times of drawups and drawdowns (see Section 1). It is shown in Landriault et al. (2015) that expected times of a drawup *<sup>δ</sup>up* and a drawdown *δdown* depend on the volatility and the trend of the drifted Brownian motion. It can be mathematically expressed as

$$\mathbb{E}[T\_{\text{up}}(\delta\_{\text{up}})] = \frac{\varepsilon^{-\frac{2\mu}{\sigma^2}\delta\_{\text{up}}} + \frac{2\mu}{\sigma^2}\delta\_{\text{up}} - 1}{\frac{2\mu^2}{\sigma^2}},\tag{10}$$

and

$$\mathbb{E}[T\_{down}(\delta\_{down})] = \frac{e^{\frac{2\mu}{\sigma^2}\delta\_{down}} - \frac{2\mu}{\sigma^2}\delta\_{down} - 1}{\frac{2\mu^2}{\sigma^2}}.\tag{11}$$

Using the Taylor expansion *e*± 2*μσ*2 *δ* = 1 ± 2*μσ*2 *δ* + ( 2*μσ*2 *δ*)<sup>2</sup> 2! + <sup>O</sup>(*μ*<sup>3</sup>) and letting *μ* → 0, one can recover that in the case with no trend the equation simplifies to

$$\mathbb{E}[T\_{up}(\delta)] = \mathbb{E}[T\_{down}(\delta)] = \frac{\delta^2}{\sigma^2}. \tag{12}$$

These equations establish a scaling law dependence between waiting times of a directional change, volatility, and the selected size of the directional-change threshold. Indeed, in the analysis of Glattfelder et al. (2011) it was empirically found that in the FX market the average waiting time is proportional to the second power of the directional-change threshold *δ* used to identify alternating trends:

$$
\langle T(\delta) \rangle \sim \delta^2. \tag{13}
$$

The closeness of Equations (12) and (13) confirms the assumption that the evolution of high-frequency prices expressed in terms of the directional-change intrinsic time has similar properties to the random walk.

### *6.2. Number of Directional Changes*

Let *<sup>N</sup>*(*<sup>δ</sup>down*; *σ*, *μ*, [0, *T*]) denote the number of drawdowns of the size *δdown* observed within the time interval [0, *T*] in Brownian motion process with parameters *μ* and *σ*. Since the sequence *Tdown*(*<sup>δ</sup>down*)1, *Tdown*(*<sup>δ</sup>down*)2,... is the sequence of non-negative, independent, and identically distributed random variables, the sequence {*ψn*; *n* ∈ N} where *ψn* = *Tdown*(*<sup>δ</sup>down*)1 + ... + *Tdown*(*<sup>δ</sup>down*)*n* + ... is the renewal point process. Thus, *<sup>N</sup>*(*<sup>δ</sup>down*; *σ*, *μ*, [0, *T*]) can be considered as the renewal counting process and its values can be found applying the Theorem 6.1.1 of Rolski et al. (2009) (Landriault et al. 2015) to the waiting time Equation (11):

$$\lim\_{T \to \infty} \mathcal{N}(\delta\_{down}; \sigma, \mu\_\star[0, T]) = \mathbb{E}[T\_{down}(\delta\_{down})]^{-1} T = \frac{T \frac{2\mu^2}{\sigma^2}}{e^{\frac{2\mu}{\sigma^2} \delta\_{down}} - \frac{2\mu}{\sigma^2} \delta\_{down} - 1}. \tag{14}$$

Correspondingly, the expected number of drawups *<sup>N</sup>*(*<sup>δ</sup>up*; *σ*, *μ*, [0, *T*]) takes the form

$$\lim\_{T \to \infty} N(\delta\_{up}; \sigma, \mu\_\prime [0, T]) = \mathbb{E} [T\_{up}(\delta\_{up})]^{-1} T = \frac{T \frac{2\mu^2}{\sigma^2}}{e^{-\frac{2\mu}{\sigma^2} \delta\_{up}} + \frac{2\mu}{\sigma^2} \delta\_{up} - 1}. \tag{15}$$

Equations (14) and (15), combined together, give the estimate of the number of directional changes consequently following each other:

$$\mathbb{E}[\mathcal{N}(\delta\_{up}, \delta\_{down}; \mu, \sigma, [0, T])] = \frac{2T \frac{2\mu^2}{\sigma^2}}{\bar{\sigma}^{-\frac{2\mu}{\sigma^2}\delta\_{up}} + \bar{\sigma}^{\frac{2\mu}{\sigma^2}\delta\_{down}} + \frac{2\mu}{\bar{\sigma}^2}(\delta\_{up} - \delta\_{down}) - 2}. \tag{16}$$

The expression is simplified in the trend-less case (*μ* → 0) to the following form:

$$\mathbb{E}[N(\delta\_{up}, \delta\_{down}; \sigma\_\prime[0, T])] = \frac{2T\sigma^2}{\delta\_{up}^2 + \delta\_{down}^2}.\tag{17}$$

The theoretical dependence of the number of directional changes and the properties of underlying process resemble the empirical observations of Guillaume et al. (1997). The authors mention there that *N*(*δ*) ∼ *δ*−<sup>2</sup> (for *δ* = *<sup>δ</sup>up* = *δdown*).

Monte Carlo statistical tests were performed to numerically verify the accuracy of Equations (10), (11) and (16). Results of the tests are provided in Table 1. We selected only positive trend values *μ* since the equations are symmetrical with respect to the direction of the trend. Values in Table 1 exhibit high similarity of both empirical and theoretical results.

The meaning behind the provided equations is that the absolute size and the ratio of directional-change thresholds used to dissect a price curve into a sequence of upward and downward trends affect the frequency and the total number of events registered within a given time interval. It follows from Equations (14) and (15) that the combination *γ* = *μσ*2 is the crucial factor affecting the expected number of intrinsic events9. We check the number of directional changes registered by a couple of thresholds in three extreme scenarios: *μσ*2 = 0 (Figure 2a), *μσ*2 0 (Figure 2b), and *μσ*2 0

<sup>9</sup> The expression *γ* is known in the insurance industry as "adjustment coefficient" or "the Lundberg exponent" (Asmussen and Albrecher 2010). It finds its application in the ruin theory dating back to 1909 (Lundberg 1909). It is also described as the optimal information theoretical betting size called Kelly Criterion (Kelly 2011).

(Figure 2c). A diverse set of dissection procedures was applied to the randomly generated time series defined by the parameters *γ*. All results were composed as a heatmap where each point corresponds to the number of directional changes observed by a pair of thresholds {*<sup>δ</sup>up*, *<sup>δ</sup>down*} (Y- and X-axis of the plots) in a time series of the given length (Figure 2).

**Table 1.** Waiting times and number of directional changes in a Monte Carlo simulation. *μ* and *σ* are parameters of the Brownian motion used for the test. There are 10<sup>9</sup> ticks in the simulated time series. *NMC DC* , *TMC up* , and *TMC down* are the numbers of directional changes and the average waiting times registered in the Monte Carlo simulation. <sup>E</sup>[*NDC*], <sup>E</sup>[*Tup*], and E[*Tdown*] are theoretical values dictated by Equations (16), (10) and (11) correspondingly. Values *<sup>σ</sup>*<sup>−</sup>*TMC up* and *<sup>σ</sup>*<sup>−</sup>*TMC down* are standard deviations of empirical and theoretical waiting times.


**Figure 2.** Heatmaps of the number of directional changes observed by the pair of directional-change thresholds {*<sup>δ</sup>up*, *<sup>δ</sup>down*} (Y- and X-axis of the plots) in a timeseries of the given length (Geometrical Brownian Motion (GBM), 10<sup>9</sup> steps in each simulation). Selected trend and volatility values: (**a**) *μ* = 0, *σ* = 0.15; (**b**) *μ* = −3, *σ* = 0.15; (**c**) *μ* = 3, *σ* = 0.15. The values on the plots coincide with the ones computed using Equation (17).

Panel 2a in Figure 2 corresponds to the set of experiments where the Brownian Motion trend is equal to zero. It follows from Equation (17) that in such conditions the value <sup>E</sup>[*N*(*<sup>δ</sup>up*, *δdown*; *σ*, [0, *T*])] should be constant along circular contours *δ*2*up* + *<sup>δ</sup>*2*down* = *δ*2 for *δ* > 0. The colour gradient in the provided picture confirms the noted dependence. It is shown in panels 2b and 2c of Figure 2 that the circular contours transform into ellipses when the "adjustment coefficient" *γ* is significantly smaller or significantly bigger than zero. This phenomenon can be interpreted in the following way: if <sup>E</sup>[*N*(*<sup>δ</sup>up* = *δdown*; *γ* = 0, [0, *T*])] is the expected number of directional changes registered in the drift-less time series of given length and characterised by the fixed *σ* then for any *γ* greater or smaller than zero there is always such a couple of non equal thresholds {*<sup>δ</sup>up*, *<sup>δ</sup>down*|*<sup>δ</sup>up* = *<sup>δ</sup>down*)} that

$$\mathbb{E}[\mathcal{N}(\delta\_{up}, \delta\_{down} \mid \delta\_{up} \neq \delta\_{down}; \gamma \neq 0, [0, T])] = \mathbb{E}[\mathcal{N}(\delta\_{up}, \delta\_{down} \mid \delta\_{up} = \delta\_{down}; \gamma = 0, [0, T])].\tag{18}$$

In other words, any process characterised by a certain degree of persistent trend could be treated as the one without the trend by tuning the size and the ratio of selected directional-change thresholds. The property is essential for risk managemen<sup>t</sup> techniques constructed on top of directional-change intrinsic time approach. An example of real application of this fact is provided in Golub et al. (2017). The authors employed asymmetric thresholds to design an optimal inventory control function sensitive to the significant price trend changes.
