Relevant Stylized Facts About Bitcoin

When building risk and valuation models for a new asset class, we need to first understand the stylized facts about its price generating process. For traditional assets, a useful reference on the topic is Cont (2001)[1], while for cryptocurrencies the studies are still incomplete and often lack academic rigour. The paper[0] we are looking at today has analyzed the BTC stylized facts and compared them to those of other financial assets, an exercise that will help in our research on what kind of risk measures to use in our digital asset investing.

Data

The data is sourced from Coinmarketcap for the daily frequency (between 28 April 2013 and 14 February 2019), and from BitcoinCharts for higher frequencies (7 January 2018 to 7 February 2018). The only pre-processing done is to take the log of the returns. It looks something like Figure 1.

Figure 1: BTC price and returns in US$ between 28 April 2013 and 14 February 2019. Source: da Cunha and da Silva (2020)[0].

Stylized facts

In Finance, a stylized fact refers to empirical findings so consistent across assets that it is generally accepted as true. Let’s list those found in this paper:

1. Fat tails: the empirical distribution of BTC returns doesn’t fit well a Gaussian, thus they are not driven by an additive process. In particular, the distribution of BTC returns is confirmed as having fat tails. Fat tails mean that dispersion measurements won’t necessarily have finite theoretical moments. Luckily in this case, the theoretical variance of the distribution of returns is shown to be finite, converging to a finite value after 2×1⁰⁴ minutes (approx. 2 weeks of data).
2. Aggregational Gaussianity: for shorter frequencies (1 to 200 minutes), the distribution of returns is leptokurtic (Kurtosis above 3), while it tends to a Gaussian with longer intervals (200 to 400 minutes). What this means is that the shape of the distribution changes with scale, and it converges to a Gaussian with increasing timescale. This shows that the same statistical methodologies applied to different frequencies might show different results, as some hypothesis would not be realistic in the case of non-Gaussian returns and would thus require some relaxation.
3. Fluctuation scaling: the second moment (variance) is a power-law of the expected return, i.e. it follows the Taylor’s law of temporal fluctuation scaling, which is present in many natural systems such as the human genome (check this paper if you want to learn more). In particular, this means that the fluctuation of activity (variance of returns) grows monotonically with the average activity (expected returns). This is an important law to consider, because it characterizes the asymptotic behavior of commonly used measures of the risk-adjusted performance, such as Sharpe or Sortino ratios.
4. Clustering volatility: using the Fourier transform of returns to estimate autocorrelations, authors find no correlation in log-returns, but they do find it in the variance. For traditional finance, this has been noted by Mandelbrot (1963)[2] and has been tackled in models such as ARCH (Engle, 1982)[3] and GARCH (Bollerslev, 1986)[4]. Related to this, we should remember that at higher frequencies log-returns do actually appear auto-correlated, most likely due to market microstructure effects. This obviously needs to be taken into account when building e.g. liquidity analysis algorithms for executions systems, as this is as close to market microstructure as it gets.
5. Positive correlation between BTC volatility and its volume: the authors used inverse Fourier transform to estimate the correlation between volume and volatility. This correlation is shown to be weak (below 0.10) on the short-term and peaking at medium-term (at 0.30), i.e. high volumes correspond to higher risks, especially in the medium-term (from 2 weeks onwards).
6. Asymmetry in time scales: the authors then coarse-grain the returns by adding 4000 minutes to the fine-scale return calculation interval (previously at 1 minute), and calculate the correlation of these returns with the fine-scale volatility. This showed asymmetry in time scales between positive and negative lags, with coarse-grained (lower frequency) measures of volatility predicting the fine-scale (higher frequency) volatility better than the other way round.

These results are important when fitting traditional finance models on cryptocurrencies because we need to understand if the same hypothesis hold. If not, these hypothesis need to be relaxed and appropriate methodologies used to avoid biased or non-convergent models.

What’s missing

There are more stylized facts the paper has not consider, and it will be a useful exercise for anyone building risk and portfolio management systems. These are the traditional finance stylized facts covered in [1], but not in the reviewed paper:

• Gain and Loss asymmetry: for Equities we see larger downwards as compared to upward movements. This is usually the case also for Commodities, but not for FX. This would help to understand how is the market treating cryptocurrencies. This per se would not be enough to define e.g. BTC as a Commodity, but in case we argue for that, it would be helpful also to show the same stylized facts as Commodities. Interestingly, some research showed that the gain-loss asymmetry can be different between mature and less mature markets (e.g. between developed and emerging markets). Check out also Kahneman and Tversky (Prospect Theory) for a potential behavioural explanation of this phenomenon.
• Intermittency (heteroscedasticity): bursts of returns volatility at any time-scale. This is present in Cont (2001)[1], but we can also argue it’s covered by the clustering volatility stylized fact, so it’s fine to not cover it here.
• Conditional heavy tails: the authors could have fit a GARCH model and checked whether errors present autocorrelation (clustering). If that’s the case, it would mean that GARCH doesn’t capture long memory in the volatility, which means we need to adjust our volatility estimates when assessing the risk of our cryptocurrency portfolios, otherwise we would be underestimating our real exposure.
• Leverage effect: this means there is a negative correlation between volatility and returns of an asset, a very well known stylized fact in traditional finance.
• Slow decay of autocorrelation in absolute returns: this is also not far from volatility clustering, so also no need to cover it here.

Conclusions

The paper adds also considerations on similarities between BTC price sand some natural phenomena (such as naturally occurring quakes), but I’ll leave this the paper if you are interested. Overall, what this particular study showed is that a good number of traditional assets stylized facts are applicable to cryptocurrency prices. The study didn’t show how these impact modelling efforts, either forecasting or risk evaluation, but that’s a good avenue for further research.

References

[0] C.R. da Cunha, R. da Silva, Relevant stylized facts about bitcoin: Fluctuations, first return probability, and natural phenomena, Physica A: Statistical Mechanics and its Applications, Volume 550, 2020, 124155, ISSN 0378–4371, https://doi.org/10.1016/j.physa.2020.124155 (arXiv).

[1] R. Cont, Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues, Quantitative Finance, 1, 223–236, 2020, http://dx.doi.org/10.1080/713665670.

[2] Mandelbrot, B. B., The Variation of Certain Speculative Prices, The Journal of Business 36, №4, (1963), 394–419.

[3] Engle, Robert F. (1982). “Autoregressive Conditional Heteroscedasticity with Estimates of Variance of United Kingdom Inflation”. Econometrica. 50 (4): 987–1008. doi:10.2307/1912773. JSTOR 1912773.

[4] Bollerslev, Tim (1986). “Generalized Autoregressive Conditional Heteroskedasticity”. Journal of Econometrics. 31 (3): 307–327. CiteSeerX 10.1.1.468.2892. doi:10.1016/0304–4076(86)90063–1.

Disclaimer

Not a financial advice, solicitation, or sale of any investment product. The information provided to you is for illustrative purposes and is not binding on Cloudwall Capital. This does not constitute financial advice or form any recommendation, or solicitation to purchase any financial product. The information should not be relied upon as a replacement from your financial advisor. You should seek advice from your independent financial advisor at all times. We do not assume any fiduciary responsibility or liability for any consequences financial or otherwise arising from the reliance on such information.

You may view this for information purposes only. Copy, distribution, or reproduction of all or any portion of this article without explicit written consent from Cloudwal is not allowed.