*6.3. Instantaneous Volatility*

The volatility size of the financial time series is an inevitable component of any financial risk analysis. Therefore, a clear understanding of the way how volatility changes over time is particularly important for risk managemen<sup>t</sup> and inventory control problems. Classical volatility estimation methods, also called "natural" or "traditional" estimators (Cho and Frees 1988)10, primarily rely on physical time as the persistent measure of the intervals when the price returns should be computed. The fact that the variance of returns on assets tends to change over time creates obstacles on the way of employing the "traditional" volatility estimators. The changing variance, also known as the stochastic volatility, became a cornerstone for multiple research works (for example, Aı and Kimmel (2007); Andersen and Lund (1997); Barndorff-Nielsen and Shephard (2002); Campbell et al. (2018) and many others).

Values, computed by "natural" estimators, dominantly correspond to the integrated volatility of the studied process. The integrated volatility describes the averaged price activity over non-zero time intervals. Alternative estimators, designed to reveal the size of the volatility as the time interval approaches zero (instantaneous volatility), are mostly based on Fourier analysis<sup>11</sup> and require extensive computation efforts (see Chapter 3 in Mancino et al. (2017)). Therefore, new methods, capable of describing the price evolution independently of the flow of the price in physical time, should be employed to overcome the existing volatility estimation difficulties.

The directional-change intrinsic time concept is by design agnostic to the speed of the price change. Risk-management tools, based on top of the concept, automatically adapt their performance to treat the changing price activity better. This property of directional-change intrinsic time, together with analytical Equations (16) and (17), bring the idea of a new volatility estimator devoid of the shortcomings of the equidistant time in finance. It follows from Equation (17) that the volatility can be estimated for a trendless time series by counting the number of directional changes within the time interval [0, *<sup>T</sup>*]:

$$
\sigma\_{\rm DC} = \delta \sqrt{\frac{N(\delta)}{T}}.\tag{19}
$$

We use the superscript *DC* to distinguish the volatility computed through the directional-change intrinsic time from volatility computed by the traditional estimators. The latter we will mark by *σtrad*.

Equation (19) solely computes the volatility part *σ* of the Brownian proces. That contrasts the "natural" volatility estimation techniques where the entire stochastic *σdWt* part is typically measured. That stochastic factor includes the noise component *dWt*. Therefore, the directional change approach employed for volatility measurements can be classified as the true estimator of the instantaneous volatility. Further, we apply Equation (19) to study changing dynamic of financial time series throughout one week. We reveal volatility seasonality patterns of three FX exchange rates, crypto market BTC/USD, and the stock index S&P500.

<sup>10</sup> The work Cho and Frees (1988) is particularly interesting due to the analysis the authors did to compare volatilities computed by "natural" and "temporal" estimators. The latter employs time intervals measured between consequent and alternating price moves of fixed relative size and thus is very close to the approach presented in the current paper.

<sup>11</sup> The type of mathematical analysis applied to identify patterns or cycles in a normalised time series data.
