**1. Introduction**

All events relevant to the performance of the financial system such as political decisions, natural disasters, or economic reports rarely happen synchronously and are typically not equally spaced in time. A sequence of them has a non-homogeneous nature and is not characterised by any vital autocorrelation function. Ultimately, the change of days and nights, as well as seasons, is dictated by the natural structure of the physical world which is barely connected to the flow of financial activity. Human minds, with the whole diversity of peculiar and indescribable characteristics, are primal engines of all market's evolutionary shifts. The global market, where the majority of transactions happen online and where traders, dealers, and market makers are distributed all around the world, is completely blind and deaf to the periodicity of days and nights, as well as to the climate factors of any standalone region of the Earth. New statistical tools, agnostic to the flow of the physical time, should be employed in order to handle the inner periodicity of the financial activity efficiently. In this work, we explore a concept of the endogenously defined time in finance applied to evaluate seasonality in markets' activity.

Probabilities of price drops and price rises between the running price maxima and running price minima are one of the most well-known risk-factors in finance. These probabilities are also called drawups and drawdowns. Numerous research works have focused on the analysis of the size, periodicity, and the time of recovery associated with drawups and drawdowns in traditional markets. The joint Laplace transform was utilised by Taylor (1975) for deriving the expected time until a new drawup in a drifted Brownian motion occured (traditionally considered as the model for historical price returns). The joint probability of observing a drawup of a given size after a drawdown, during a given term, was analysed as a homogeneous diffusion process in Pospisil et al. (2009). Zhang (2015) derived the joint probability in the context of exponential time horizons (the horizons are exponentially distributed random variables). The authors also described the law of occupation times for both drawup and drawdown processes. These and other theoretical findings connected to the price trend reversals were successfully applied to real financial problems such as studding market crashes.

Market crashes, pronounced in the abnormal price decreases, might severely impact the long-term stability of markets. It is especially important to estimate the probability of the next crash occurrence within a given period of time. Many research works were done on studying the crash probabilities using the normal distribution of price returns as the proxy for the real process. However, extreme price drops occur more often in the real world than what should happen when the distribution of returns coincides with the normal one. Fat-tailed distributions of returns ground the observed phenomenon. The distributions were discovered in the stock market (Jondeau and Rockinger 2003; Koning et al. 2018; Rachev et al. 2005), in the Forex (FX) (Cotter 2005; Dacorogna et al. 2001), as well as in Bitcoin, prices (Beguši´c et al. 2018; Liu et al. 2017). The fat tails, accompanied by the extensive discontinuity of the price curve (jumps), make the equally spaced time intervals inconvenient for high-frequency market analysis. Research tools, capable of working independently to the price distribution, should be called to deal with the erratic price evolution. Prices, at which drawdowns and drawups of the given size are registered, are independent of the time component of the price progression. Thus, the drawdown and drawups are the concepts especially useful of handling the dynamics of high-frequency markets. A sequence of drawdowns and drawups, following each other, can describe the evolution of a time series purely from the price point of view. The efficient set of forecasting techniques aimed at identifying appropriate conditions for future market crashes should inevitably be supplied by risk-management tools managing sequences of drawdowns and drawups.

In this research work, we investigate the connection between the observed number of alternating drawdowns and drawups (directional-change intrinsic time measure) and the instantaneous volatility. Non-parametric estimation of instantaneous volatility is still a relatively new topic which, to the extent of our knowledge, has not been studied before from the point of view of directional-change intrinsic time. Obtained in the work, analytical expressions are employed to reveal the seasonality structure of instantaneous volatility typical for high-frequency exchange rates. The described tools and experiments contribute to the collection of existing literature on directional-change intrinsic time and the seasonality properties of high-frequency markets. The tools will benefit high-frequency traders whose computer algorithms primarily operate on ultra-short time intervals where the short-term properties dominate over the long-term statistical characteristics (Gençay et al. 2001; Hasbrouck 2018).

Three distinctive markets were considered in the work: FX (EUR/USD, EUR/JPY, and EUR/GBP), stocks (S&P500), and crypto (BTC/USD). All experiments are performed on the time series of the highest granularity: tick-by-tick data. That high granularity is essential considering the substantially growing interest in high-frequency trading after the 2008 financial crisis (Kaya et al. 2016). The data corresponds to the recent time period from 2011 to 2018 and is obtained from the largest trading venues (*JForex* and *Kraken*) opened for traders of any size. Each of the time series used in the empirical analysis is at least four years long. Such an extended length allows us to claim that properties specific for any particular period of time should not be pronounced in the obtained results.

The outline of the remaining paper is as follows. Section 2 provides a brief overview of research works on the properties of drawups and drawdowns. Section 3 gives detailed reasoning on the need for directional-change intrinsic time and describes a set of literature where the concept was successfully applied. Existing studies on the volatility seasonality of high-frequency markets is provided in Section 4. Section 5 describes the data used in the experiments and Section 6 outlines how the number of directional changes is connected to the instantaneous volatility. In Section 7 we present all results obtained by the traditional, as well as the novel, volatility measurement techniques and also describe

the application of theta time concept aimed to minimise the seasonality pattern. Section 8 concludes the main body of the paper and proposes the potential use of the developed technique. Appendix A concludes the paper by presenting a set of experiments where the comparison of considered markets seasonality patterns is presented.

### **2. Drawdowns and Drawups: An Introduction**

Probabilities of financial drawdowns and drawups were extensively studied and presented in multiple seminal research works. Drawdowns of extensive size are usually associated with market crashes. Carr et al. (2011) proposed a new insurance technique aimed to protect investors against unexpected price moves. The authors also covered a novel way of hedging liabilities associated with these risks. Zhang and Hadjiliadis (2012) employed statistical properties of drawdowns as an estimate of the stock default risk and also provided a risk-management mechanism affecting the investor's optimal cancellation timing. In Schuhmacher and Eling (2011) drawdowns are considered as one of 14 reward-to-risk ratios alternative to the widely known performance measures such as the Sharpe ratio. In Grossman and Zhou (1993) and Chekhlov et al. (2005) the properties of drawdowns were also applied as an estimate of the portfolio optimisation problem. The latter can be personalised to match traders' or investors' expectations and their tolerance to the size and the length of the market disruption.

Drawdowns *Dt* and drawups *Ut*, also called rallies in Hadjiliadis and Veˇceˇr (2006), registered by the moment of time *t*, depend on the running price maxima *St* and the running price minima *St* (Dassios and Lim 2018; Landriault et al. 2015; Mijatovi´c and Pistorius 2012; Zhang and Hadjiliadis 2012). These reference points hinge on the set of historical prices *Ss* and are mathematically defined in the following way:

$$\bar{S\_l} = \sup \{ S\_\circ : 0 \le s \le t \} \quad \text{and} \quad \underline{S\_l} = \inf \{ S\_\circ : 0 \le s \le t \}, \tag{1}$$

where *t* ≥ 0 and the interval [0, *t*] is fixed. Drawdowns and drawups are the differences between the final price of the given time interval *St* and the registered local maxima and minima:

$$D\_l = \overline{S\_l} - S\_l \quad \text{and} \quad lI\_l = S\_l - \underline{S\_l}. \tag{2}$$

The waiting time *τDa* becomes measured once a price curve experiences a drawdown *Dt* of the size *a*. Similarly, *τUa* is the waiting time associated with a drawup of the size *a*. In details:

$$
\tau\_a^D = \inf\{t \ge 0 : D\_t \ge a\} \quad \text{and} \quad \tau\_a^{\mathcal{U}} = \inf\{t \ge 0 : \mathcal{U}\_l \ge a\}.\tag{3}
$$

The waiting time *τa* measures the period of physical time which elapses before the first drawdown (potentially interpreted as a market crash) becomes registered.
