# Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time

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## Abstract

**:**

## 1. Introduction

## 2. Drawdowns and Drawups: An Introduction

## 3. Directional-Change Intrinsic Time

## 4. Seasonality

#### 4.1. Traditional Markets

#### 4.2. Bitcoin Seasonality

## 5. Data

#### Inner Price

## 6. Methods

#### 6.1. Waiting Time

#### 6.2. Number of Directional Changes

#### 6.3. Instantaneous Volatility

## 7. Results

#### 7.1. Number of Directional Changes

#### 7.2. Realised versus Instantaneous Volatility

^{−3}, correspondingly. Putting these coefficients into Equation (21), one can calculate that thresholds reciprocal to the selected time intervals $\Delta {t}_{1},\dots ,\Delta {t}_{4}$ are: $\delta (\Delta {t}_{1})=0.013\%$, $\delta (\Delta {t}_{2})=0.039\%$, $\delta (\Delta {t}_{3})=0.095\%$, and $\delta (\Delta {t}_{4})=0.458\%$. It is worth mentioning that applied scaling parameters are relevant only to the FX market which was the object of the research in Glattfelder et al. (2011). To the extent of our knowledge, parameters specific to Bitcoin prices, as well as to the S&P500 index, were not mentioned in the scientific literature before. Therefore, as the first step, we obtained the parameters by studying the “time of total-move” scaling law of historical Bitcoin, and SPX500 returns. The log-log plot of waiting times ${T}_{TM}\left(\delta \right)$ versus the directional-change threshold size $\delta $ is provided in Figure 4. The red line marks BTC/USD scaling law and is shown together with black, yellow, and green lines computed for EUR/USD, SPX500, and Geometrical Brownian Motion (GBM) correspondingly. Settings of the latter are chosen to mimic returns typical for the FX market.

^{−4}, 9.94 × 10

^{−3}, and 4.60 × 10

^{−4}, correspondingly. These values are significantly different due to the unlike scale of the corresponding volatility. This volatility dependent scaling parameter is not critical for the current analysis and will be discussed in the future research works.

#### 7.3. Discrete Price Effect

#### 7.4. Volatility Seasonality

#### 7.5. Volatility Autocorrelation and Theta Time

## 8. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Daily Seasonality

**Figure A1.**Daily realised volatility ratio of BTC/USD measured over 10-min time intervals dissecting the entire week into 1008 bins. The volatility is computed according to the “traditional” approach (Equation (20)).

**Figure A2.**Daily realised volatility ratio of the given exchange rates (labelled by ***) and EUR/USD. The volatility is computed according to the “traditional” approach (Equation (20)).

**Figure A3.**Daily instantaneous volatility ratio of BTC/USD measured over 10-min time intervals dissecting the entire week into 1008 bins. The volatility is computed according to the novel approach (Equation (19)).

**Figure A4.**Daily instantaneous volatility ratio of the given exchange rates (labelled by ***) and EUR/USD. The volatility is computed according to the novel approach (Equation (19)).

**Table A1.**Daily instantaneous volatility ratio and the weekly standard deviation of two FX (EUR/JPY and EUR/GBP), one crypto (BTC/USD), and one stock (SPX500) exchange rates to EUR/USD. Columns $Rati{o}_{*}$ stands for the ratio of the average daily volatility of the corresponding exchange rate and one of EUR/USD. The average value computed over 10-min intervals. Columns $st{d}_{*}$ contain the standard deviation values of the volatility ratio over the set of 10-min intervals. The subscripts $trad$ and $DC$ label the measures made using the traditional volatility estimator (Equation (20)) and the novel approach (Equation (19)) correspondingly. The daily volatility ratios are graphically presented in Figure A2 and Figure A3.

${\mathit{Ratio}}_{\mathit{trad}}$ | ${\mathit{std}}_{\mathit{trad}}$ | ${\mathit{Ratio}}_{\mathit{DC}}$ | ${\mathit{std}}_{\mathit{DC}}$ | ${\mathit{Ratio}}_{\mathit{trad}}/{\mathit{Ratio}}_{\mathit{DC}}$ | ${\mathit{std}}_{\mathit{trad}}/{\mathit{std}}_{\mathit{DC}}$ | ||
---|---|---|---|---|---|---|---|

Monday | EUR/JPY | 1.35 | 0.27 | 1.03 | 0.12 | 1.31 | 2.25 |

EUR/GBP | 0.89 | 0.1 | 0.64 | 0.04 | 1.39 | 2.50 | |

BTC/USD | 11.75 | 3.99 | 1.35 | 0.51 | 8.70 | 7.82 | |

SPX500 | 0.97 | 0.51 | 0.57 | 0.29 | 1.70 | 1.76 | |

Tuesday | EUR/JPY | 1.35 | 0.23 | 1.02 | 0.11 | 1.32 | 2.09 |

EUR/GBP | 0.88 | 0.13 | 0.63 | 0.05 | 1.40 | 2.60 | |

BTC/USD | 11.7 | 4.53 | 1.35 | 0.52 | 8.67 | 8.71 | |

SPX500 | 0.95 | 0.47 | 0.57 | 0.3 | 1.67 | 1.57 | |

Wednesday | EUR/JPY | 1.31 | 0.3 | 1.02 | 0.12 | 1.28 | 2.50 |

EUR/GBP | 0.88 | 0.19 | 0.64 | 0.05 | 1.38 | 3.80 | |

BTC/USD | 10.83 | 3.74 | 1.29 | 0.46 | 8.40 | 8.13 | |

SPX500 | 0.93 | 0.42 | 0.53 | 0.26 | 1.75 | 1.62 | |

Thursday | EUR/JPY | 1.34 | 0.22 | 1.03 | 0.12 | 1.30 | 1.83 |

EUR/GBP | 0.89 | 0.18 | 0.64 | 0.05 | 1.39 | 3.60 | |

BTC/USD | 11.42 | 4.42 | 1.29 | 0.5 | 8.85 | 8.84 | |

SPX500 | 0.93 | 0.49 | 0.57 | 0.28 | 1.63 | 1.75 | |

Friday | EUR/JPY | 1.35 | 0.28 | 1.03 | 0.12 | 1.31 | 2.33 |

EUR/GBP | 0.9 | 0.17 | 0.64 | 0.06 | 1.41 | 2.83 | |

BTC/USD | 11.81 | 4.93 | 1.3 | 0.54 | 9.08 | 9.13 | |

SPX500 | 0.98 | 0.46 | 0.61 | 0.31 | 1.61 | 1.48 |

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1. | A basic polynomial functional relationship where a change in input results in a proportional change in output. |

2. | A growing list of records containing information on the ownership of all existing Bitcoins. |

3. | Information on all cryptocurrencies and trading venues can be found at Coinmarketcap.com. |

4. | At the moment of writing the paper, Wall Street and other big financial hubs are considering trading cryptocurrencies, which will potentially result in the higher segregation level. |

5. | According to the Bank for International Settlements the daily average FX trading volume in April 2016 was $5.1 trillion (BIS 2016) when the highest registered volume in the crypto market is to the date only $45.8 billion (https://coinmarketcap.com/charts/). |

6. | |

7. | |

8. | |

9. | The expression $\gamma $ 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). |

10. | 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. |

11. | The type of mathematical analysis applied to identify patterns or cycles in a normalised time series data. |

12. | It had a minimum at $230 per Bitcoin, temporary maximum at about $20,000, and then a drop to $6000. |

13. | The evidence that the distribution of returns approaches the normal one measured over longer timescales. |

14. | According to the Table 2. |

**Figure 1.**A part of EUR/USD price curve (grey) dissected into a set of directional-changes (grey squares) using a directional-change threshold $\delta $. 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 $\delta $. 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 ${T}_{1}\left(\delta \right),\dots ,{T}_{6}\left(\delta \right)$.

**Figure 2.**Heatmaps of the number of directional changes observed by the pair of directional-change thresholds $\{{\delta}_{up},{\delta}_{down}\}$ (Y- and X-axis of the plots) in a timeseries of the given length (Geometrical Brownian Motion (GBM), ${10}^{9}$ steps in each simulation). Selected trend and volatility values: (

**a**) $\mu =0$, $\sigma =0.15$; (

**b**) $\mu =-3$, $\sigma =0.15$; (

**c**) $\mu =3$, $\sigma =0.15$. The values on the plots coincide with the ones computed using Equation (17).

**Figure 3.**Heat map of the number of directional changes calculated in (

**a**) EUR/USD, (

**b**) BTC/USD, and (

**c**) SPX500 time series. Each point on the grid represents the number of directional changes registered by a unique pair of thresholds $\{{\delta}_{up},{\delta}_{down}\}$. Heatmaps have different scales. Yellow solid lines, specific for each heatmap, label the examples of the areas along which the number of intrinsic events is constant. The dashed lines represent the theoretical areas of the equal number of intrinsic events observed in case of the trend-less time series. White dashed lines are parts of circles centred around the left bottom corner of each picture. The lines go through the intersection of the solid yellow lines and the diagonal of each picture.

**Figure 4.**Time of total-move scaling laws computed for BTC/USD, EUR/USD, SPX500, and Geometrical Brownian Motion (GBM). GBM’s parameters are ${S}_{0}=1.3367$, $\mu =0$, $\sigma =20\%$, $T=1$ year, and 10 million ticks in total. Scaling parameters C and E correspond to the coefficients of Equation (21).

**Figure 5.**Instantaneous volatility of three time series generated by GBM with various tick frequencies and fixed volatility ($15\%$). The volatility is computed by the directional-change approach (Equation (19)). Sizes of the directional change thresholds, used to calculate the volatility values, are put on the X-axis. Red dashed line marks the $15\%$ level.

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

**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.

**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 $\delta =0.01\%$ was used in both experiments. Each discrete time interval (bin) is $T=10$ min. There are 1008 bins in total.

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

**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: $\delta =\left(\right)open="\{"\; close="\}">0.20\%,0.10\%,0.03\%,0.01\%$. 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.

**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 $\delta =0.01\%$.

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

**Table 1.**Waiting times and number of directional changes in a Monte Carlo simulation. $\mu $ and $\sigma $ are parameters of the Brownian motion used for the test. There are ${10}^{9}$ ticks in the simulated time series. ${N}_{DC}^{MC}$, $\langle {T}_{up}^{MC}\rangle $, and $\langle {T}_{down}^{MC}\rangle $ are the numbers of directional changes and the average waiting times registered in the Monte Carlo simulation. $\mathbb{E}\left[{N}_{DC}\right]$, $\mathbb{E}\left[{T}_{up}\right]$, and $\mathbb{E}\left[{T}_{down}\right]$ are theoretical values dictated by Equations (16), (10) and (11) correspondingly. Values ${\sigma}_{{T}_{up}^{MC}}^{-}$ and ${\sigma}_{{T}_{down}^{MC}}^{-}$ are standard deviations of empirical and theoretical waiting times.

$\mathit{\mu}$, % | $\mathit{\sigma}$, % | ${\mathit{N}}_{\mathit{DC}}^{\mathit{MC}}/\mathbb{E}\left[{\mathit{N}}_{\mathit{DC}}\right]$ | $\langle {\mathit{T}}_{\mathit{up}}^{\mathit{MC}}\rangle /\mathbb{E}\left[{\mathit{T}}_{\mathit{up}}\right]$ | ${\mathit{\sigma}}_{{\mathit{T}}_{\mathit{up}}^{\mathit{MC}}}^{-}$ | $\langle {\mathit{T}}_{\mathit{down}}^{\mathit{MC}}\rangle /\mathbb{E}\left[{\mathit{T}}_{\mathit{down}}\right]$ | ${\mathit{\sigma}}_{{\mathit{T}}_{\mathit{down}}^{\mathit{MC}}}^{-}$ |
---|---|---|---|---|---|---|

1 | 10 | 1.028 | 0.968 | 2.54 × 10^{−5} | 1.019 | 2.53 × 10^{−6} |

20 | 1.009 | 0.989 | 2.78 × 10^{−6} | 1.012 | 3.32 × 10^{−7} | |

30 | 1.001 | 0.995 | 8.79 × 10^{−7} | 1.033 | 9.58 × 10^{−8} | |

6 | 10 | 1.021 | 0.971 | 2.29 × 10^{−5} | 1.043 | 2.59 × 10^{−6} |

20 | 1.005 | 0.993 | 2.94 × 10^{−6} | 1.019 | 3.29 × 10^{−7} | |

30 | 0.987 | 1.011 | 8.84 × 10^{−7} | 1.034 | 9.98 × 10^{−8} | |

11 | 10 | 1.029 | 0.968 | 2.20 × 10^{−5} | 1.011 | 2.78 × 10^{−6} |

20 | 0.994 | 1.006 | 2.72 × 10^{−6} | 0.997 | 3.30 × 10^{−7} | |

30 | 0.986 | 1.014 | 8.82 × 10^{−7} | 1.017 | 1.02 × 10^{−7} |

**Table 2.**Volatility of the considered time series computed using the “traditional” (Equation (20)) and the directional-change (Equation (19)) approaches. Provided values $\langle {\sigma}_{trad}\rangle $ and $\langle {\sigma}_{DC}\rangle $ are the average of four measurements performed with specific parameters: in the “traditional” case time intervals between observations ${S}_{n}$ and ${S}_{n-1}$ are $\Delta {t}_{1}=$ 1 min, $\Delta {t}_{2}=$ 10 min, $\Delta {t}_{3}=$ 1 h, and $\Delta {t}_{4}=$ 1 day. In the case of the directional-change intrinsic time approach, the thresholds $\delta $ are $\delta (\Delta {t}_{1})=0.013\%$, $\delta (\Delta {t}_{2})=0.039\%$, $\delta (\Delta {t}_{3})=0.095\%$, $\delta (\Delta {t}_{4})=0.458\%$ (FX prices), $\delta (\Delta {t}_{1})=0.09\%$, $\delta (\Delta {t}_{2})=0.33\%$, $\delta (\Delta {t}_{3})=0.89\%$, $\delta (\Delta {t}_{4})=5.13\%$ (BTC prices), and $\delta (\Delta {t}_{1})=0.006\%$, $\delta (\Delta {t}_{2})=0.025\%$, $\delta (\Delta {t}_{3})=0.075\%$, $\delta (\Delta {t}_{4})=0.545\%$ (SPX500).

Name | $\langle {\mathit{\sigma}}_{\mathit{trad}}\rangle ,\%$ | ${\mathit{\sigma}}_{\mathit{trad}}^{-}$ | $\langle {\mathit{\sigma}}_{\mathit{DC}}\rangle ,\%$ | ${\mathit{\sigma}}_{\mathit{DC}}^{-}$ | $\langle {\mathit{\sigma}}_{\mathit{trad}}\rangle /\langle {\mathit{\sigma}}_{\mathit{DC}}\rangle $ | ${\mathit{\sigma}}_{\mathit{trad}}^{-}/{\mathit{\sigma}}_{\mathit{DC}}^{-}$ |
---|---|---|---|---|---|---|

EUR/USD | 9.72 | 0.03 | 7.53 | 1.38 | 1.29 | 0.02 |

EUR/JPY | 11.93 | 0.12 | 8.55 | 2.07 | 1.40 | 0.06 |

EUR/GBP | 8.04 | 0.23 | 5.81 | 1.43 | 1.38 | 0.16 |

BTC/USD | 84.76 | 8.67 | 80.87 | 22.21 | 1.05 | 0.39 |

SPX500 | 13.19 | 0.67 | 6.63 | 3.24 | 1.99 | 0.21 |

**Table 3.**Parameters of the logarithmic decay $y={A}_{ACF}logx+{B}_{ACF}$ used to fit the autocorrelation function (ACF) of the number of directional changes in $\mathsf{\Theta}$-time (Figure 14).

Name | ${\mathit{A}}_{\mathit{ACF}}$ | ${\mathit{B}}_{\mathit{ACF}}$ |
---|---|---|

BTC/USD | −0.029 | 0.84 |

EUR/USD | −0.021 | 0.75 |

EUR/JPY | −0.018 | 0.65 |

EUR/GBP | −0.015 | 0.59 |

SPX500 | −0.054 | 0.69 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Petrov, V.; Golub, A.; Olsen, R.
Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time. *J. Risk Financial Manag.* **2019**, *12*, 54.
https://doi.org/10.3390/jrfm12020054

**AMA Style**

Petrov V, Golub A, Olsen R.
Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time. *Journal of Risk and Financial Management*. 2019; 12(2):54.
https://doi.org/10.3390/jrfm12020054

**Chicago/Turabian Style**

Petrov, Vladimir, Anton Golub, and Richard Olsen.
2019. "Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time" *Journal of Risk and Financial Management* 12, no. 2: 54.
https://doi.org/10.3390/jrfm12020054