3.1.2. High-Frequency Bitcoin Price Data

Multifractality tests are performed on the high-frequency minute-by-minute data of Bitcoin open prices from 22 May 2018 14:00 to 1 March 2019 11:00 with 406,089 observations. Its mean-centered log-prices and log-returns are displayed in Figure 8. Similar to other financial assets, a lack of correlation in the return increments but high correlation in the absolute returns are found. However, compared with the daily open Bitcoin price data, the high-frequency Bitcoin data display stronger auto-correlation in the absolute return increments.

**Figure 8.** High frequency mean-centered log Bitcoin open price process (**top**) and mean-centered log returns process (**bottom**).

This data set covers approximately a one year period from May 2018 to March 2019. The two corresponding annual breakdown tests for daily Bitcoin open prices covering this period in Appendix B.1 in Appendix B do not indicate multifractal scaling, despite some evidence in Tables 1 and 3 for multifractality in daily data, generally.

The analysis of the high-frequency minute-by-minute data are broadly consistent with this result. With the estimated scaling function *τ*<sup>ˆ</sup>(*q*) shown in Figure 9 and the test results in Table 4, there appears to be little evidence of multifractal scaling for the high frequency data. Nevertheless, as shown in Appendix B.2, multifractality can be detected in some cases if we partition the data into smaller subperiods.

Given these contradictory results and that the high-frequency data only spans one year, we are reluctant to infer at this stage that the Bitcoin time series scales in the same way over longer or shorter time intervals.

Overall, it is possible that multifractal scaling exists for Bitcoin prices; however, a longer time series is necessary to substantiate the use of a multifractal model and further tests are needed to discern between monofractal models as a parsimonious alternative. In the following section, we test for the presence of multifractal scaling in other financial assets. These assets have more established markets with a long history of price data. It is thus useful to compare their scaling properties to that of the Bitcoin and to identify any similarities.

**Table 4.** Hypothesis test results on high frequency Bitcoin open prices (22 May 2018–1 March 2019).


**Figure 9. Left**—Autocorrelations of mean-centered log returns (black) and absolute returns (grey) for high frequency Bitcoin open prices (22 May 2018–1 March 2019). **Right**—Scaling function for high frequency Bitcoin open prices (22 May 2018–01 March 2019).

#### *3.2. Bitcoin Compared to Other Financial Assets*

Cryptocurrencies are a new financial asset whose value depends upon the evolution of its underlying technology as well as the design of its economic model. Bitcoin in particular appears to have characteristics that span a commodity, medium of exchange and technology. As a result, it is unclear which asset class Bitcoin most resembles. In the previous section, we identified that Bitcoin's price dynamics could be characterised by multifractal scaling laws after accounting for heavy tails. In this section, we compare Bitcoin's scaling properties to other financial assets. Included in our set of comparable assets are stock indices, a foreign exchange rate series and gold futures. The S&P500 index<sup>7</sup> has been chosen to represent a large class of global equities; the Nasdaq Composite Index<sup>8</sup> illustrates the price behaviour of technology-stocks; the USD/JPY exchange rate<sup>9</sup> represents foreign currency; and gold futures prices<sup>10</sup> are also included in the comparison, as Bitcoin is often referred to as the digital version of gold.

The results of the hypothesis tests are summarised in Tables 5–7 and the associated scaling functions are depicted in Figures 10–12. Table 5 compares Bitcoin daily prices to other assets, over a short time period spanning Bitcoin's limited but available price history, from 2013–2019. We observe that the local and global results can produce conflicting evidence. The local test for the S&P500 marginally accepts the null hypothesis, where the global test result marginally accepts the alternative in favour of multifractality for BTC Daily. The global test statistic reveals that the greatest evidence for multifractal scaling is for the USD/JPY exchange series. The scaling functions for each asset class look visibly concave for all assets except the S&P 500; however, multifractality in the presence of heavy tails is only indicated for Bitcoin and the the USD/JPY, indicating on prima facie evidence that Bitcoin could share similar scaling properties to foreign exchange. However, Tables 6 and 7 reveal more compelling evidence given the larger data set spanning longer time periods. The lengthy time period is able to capture scaling behaviour for a wide range of time scales, as is necessary to establish a multifractal scaling relationship for an arbitrary set of rescaling factors. The results in Tables 6 and 7 indicate that both the NASDAQ and the USD/JPY foreign exchange series could share multifractal scaling properties along with Bitcoin. The scaling function of the Gold Futures series looks visibly concave; however, the hypothesis test accepts the null in the presence of heavy tails indicating that the concavity is a result of heavy tails as opposed to multifractal scaling.

<sup>7</sup> Retrieved from Yahoo Finance (2019) for the period 30 December 1927 to 26 February 2020 with 23,147 observations.

<sup>8</sup> Retrieved from Yahoo Finance (2020) for the period 5 February 1971 to 25 February 2020 with 12,372 observations.

<sup>9</sup> Retrieved from investing.com Australia (2020b) for the period 4 March 1988 to 28 February 2020 with 8337 observations.

<sup>10</sup> Retrieved from investing.com Australia (2020a) for the period 27 December 1979 to 28 February 2020 with 10,190 observations.


**Table 5.** Test results among different financial assets (28 April 2013–3 September 2019).

**Figure 10.** Scaling functions for S&P500, NASDAQ Index, USD/JPY Exchange and Gold Futures Prices the period 28 April 2013 to 3 September 2019.

**Table 6.** Test results among different financial assets during the dot-com bubble (3 January 1994–8 October 2004).


**Figure 11.** Scaling functions for S&P500, NASDAQ Index, USD/JPY Exchange and Gold Futures Prices during the dot-com bubble (3 January 1994–8 October 2004). The time period from 3 January 1994 to 8 October 2004 corresponds to the dot-com bubble and the dot-com crash.

**Figure 12.** Scaling functions for S&P500, NASDAQ Index, USD/JPY Exchange and Gold Futures Prices the period 4 March 1988 to 3 September 2019.


**Table 7.** Test results among different financial assets (4 March 1988–3 September 2019).
