Volatility is measured in various ways. It can be measured using standard deviations of time series data or it can be analysed and visualised using a histogram method.
Realised volatility is another way of calculating volatility and will be discussed. Finally the ARCH (Autoregressive Conditional Heteroskedasticity) models will be discussed as a way to model and analyse volatility.
3.6.1 Standard deviation
The standard deviation of a time-series is a basic statistic to measure volatility over an entire period. The standard deviation is shown by the formula below (Wegner, 2016).
ππ‘ππππππ π·ππ£πππ‘πππ = β|π₯ β π₯Μ |
π
Where:
π₯Μ = Mean of the series π = Number of observations
While the standard deviation is relatively easy to calculate, it has complex assumptions which in turn raise concerns around its accuracy and validity as a measure of risk and volatility. For the standard deviation to be used to test and measure volatility it has to be assumed that the data is normally distributed (Wegner, 2016). There are at least three reasons why time series data may not be normally distributed.
29 The first reason is that time series data like investments and Bitcoin specifically are typically skewed. Investment returns are typically asymmetrical because there are time periods of superior or poor performance (Chu et al., 2017).
The second reason is that time series data for Bitcoin typically show leptokurtosis, which means that the tails of the return distribution are typically fat. This indicates that there are a significant number of outliers in the returns of the investment (Selmi et al., 2018).
Lastly heteroskedasticity is a cause for concern. Heteroskedasticity is defined as a property of a time series where the variance of the sample data varies over time. Itβs also referred to as volatility clustering which typically happens during highly volatile times like bubbles, for which Bitcoin is known (Chu et al., 2017).
These three reasons result in the data not being normally distributed and therefore standard deviation alone is not a reliable and valid measure of volatility.
3.6.2 Histogram using the historical returns performance
An easier and more accurate method of measuring and visualising risk is by graphing a histogram with the proportion of return observations that fell within a specific range.
This allows the reader to determine the percentage of times when the investment had returns within a specific range.
Using the histogram method instead of the standard deviation method has the following advantages:
- The historical histogram method doesnβt require the returns data to be normally distributed.
- The impact of skewness and kurtosis is captured in the histogram.
These advantages provide observers with the required information to inspect the volatility.
Disadvantages of the historical histogram method, however, is that it does not account for the dataβs heteroskedasticity and does not allow statistical hypotheses testing.
3.6.3 Historical realised volatility method
The historical (realised) volatility is calculated as below (RealVol, 2018)
πππππ‘ππππ‘π¦ = 100 Γ 252
π π
30 252 is a conventional constant for trading days in a year. Even though Bitcoin trades 365 days a year, we use the standard 252 days used in finance literature, which enables comparisons with traditional currencies. t is the trading day counter and n is the total days in the time period. Rt is the continuous daily compounded return.
Rt is calculated as below (RealVol, 2018) π = ln π
π
Ln is the Natural logarithm and Pt is the underlying Bitcoin closing price at day t. Pt-1 is the underlying Bitcoin closing price the previous day.
Returns are logged and differenced to account for the auto-correlation typically associated with non-stationary time-series data. Although the realised historical volatility method above also accounts for skewness and kurtosis it does not account for heteroskedasticity as mentioned.
3.6.4 ARCH methods
As all the previous methods do not account for volatility clustering (heteroskedasticity) in time-series data, ARCH (Autoregressive Conditional Heteroskedasticity) models have been widely adopted to model volatility in recent decades (Peng et al., 2018).
The GARCH models have been a valuable tool since it was introduced by the seminal work of Engle (1982). The GARCH models are based on the ARCH models for which Engle received the Nobel Memorial Prize in Economic Sciences in 2003. ARCH models use high order lagged disturbance to properly model the behaviour of conditional variance. Marcucci (2005) agrees that the GARCH model is a parsimonious model specifically equipped to handle time series data on volatility.
Takaishi (2018) confirms that Bitcoin shows similar time series properties than other time series assets, like fat-tailed return distributions, serial correlation in returns and volatility clustering, where GARCH models are appropriate.
Various GARCH-type models exist (300+), for example the linear GARCH, the Threshold GARCH (TGARCH) by Dyhrberg (2016b), the Exponential GARCH (EGARCH) by Dyhrberg (2016a), and the Component with Multiple Threshold-GARCH (CMTGARCH) (Bouoiyour & Selmi, 2016).
Katsiampa (2017) found the optimal fitting model for analysing Bitcoin data specifically to be the AR-CGARCH model. However Charles and DarnΓ© (2018) replicated his study and found partially different results.
31 Meanwhile Chu et al. (2017) found that the IGARCH and GJRGARCH models are the optimal fit for modelling returns in the majority of large cryptocurrencies. Mensi, Al- Yahyaee and Kang (2018) on the other hand found that the FIGARCH model with structural break variables provides relatively increased forecasting accuracy.
There is accordingly no consensus in current literature on the best ARCH model to use for Bitcoin volatility modelling, especially since Bitcoin is prone to structural breaks and literature tests have been done during different time periods. This research therefore attempted to fit the best available model to the data in the time available and found that the ARCH model with a lag of 1 year was appropriate, shown by the formula below.
π΄π πΆπ»(1) = π = πΌ + πΌ π
π denotes the error terms and π is the variance.
Long memory and structural breaks in the time series data are also two important elements in modelling. If a market exhibits long memory, yesterdayβs news events still influence todayβs volatility and prices. Structural breaks are defined as unexpected shifts in a time series that invalidates existing models and can lead to forecasting errors (Preuss, Puchstein & Dette, 2015).
Another phenomenon that can sometimes be observed in time-series data are the leverage effect defined as the tendency of rising asset prices to be accompanied by declining volatility (Bouchaud, Matacz & Potters, 2001).
Although GARCH models are currently the most widely used model for Bitcoin volatility modelling, as established above, its limitation is that it does not model correctly for asymmetry and nonlinearity in the volatility data.
3.6.5 Other volatility measurements
Some other less popular methods are the Parkinson volatility method which uses the high and low values in a particular period and the Rogers and Satchell method which uses first, high, low and end of period prices. The Support Vector Regression (SVR) model, have been used by Gavrishchaka and Ganguli (2003). Other studies employ neural networks, machine learning or technical analysis (Pichl & Kaizoji, 2017).