Statistical Analysis of the Exchange Rate of Bitcoin
Analyzed the data: JC SN SC. Contributed reagents/materials/analysis instruments: JC SN SC. Wrote the paper: JC SN SC.
Bitcoin, the very first electronic payment system, is becoming a popular currency. We provide a statistical analysis of the log-returns of the exchange rate of Bitcoin versus the United States Dollar. Fifteen of the most popular parametric distributions in finance are fitted to the log-returns. The generalized hyperbolic distribution is shown to give the best fit. Predictions are given for future values of the exchange rate.
Introduced and very first documented by Satoshi Nakamoto in 2009, Bitcoin is a form of cryptocurrency—an “electronic payment system based on cryptographic proof” , instead of traditional trust.  noted that buying and selling online has become reliant “almost exclusively on financial institutions serving as trusted third parties to process electronic payments”. In other words, payments for online transactions must go through a company, such as a bank or credit card issuer, to be checked for factors such as fraud and successful payment. This kind of system is based on trust, however these checks come at a price in the form of enhanced transaction costs , meaning that we often see limitations in the form of minimum spend thresholds for electronic payments—i.e., on credit or debit cards. Bitcoin transactions are non-reversible—they are “computationally impractical to switch sides”  and can help to reduce fraud.
Interest in Bitcoin has grown at an enlargening rhythm in latest years. At the end of August 2013, the total available Bitcoins were valued at over 1.Five billion United Stated Dollars (USDs), and in December two thousand thirteen the processing power of the Bitcoin network was claimed to be “harshly three hundred times the combined power of the top five hundred supercomputers” [Two]. [Trio] states that this is because supporters of Bitcoin see it as “an ideal currency for mainstream consumers and merchants”. In brief, the high liquidity, diminished costs and the high speed of Bitcoin’s partially anonymous system are what make this currency so interesting [Three].
From a broader perspective, Bitcoin is not presently managed by a central governing figure, reducing privacy concerns. In addition, Bitcoin is not linked with any type of commodity, for example, gold or silver [Four]. Due to the decentralised nature of Bitcoin, the network is instead managed by its users. The Bitcoin system utilises a peer-to-peer network of all those who are involved in creating and trading Bitcoins, to process and check all transactions. Therefore “each participant is obliged to maintain the entire transaction history of the system rendering all transactions translucent” [Four]. This in theory should create an incentive for all users to protect the Bitcoin network. The freedom of Bitcoin may also permit organisations such as WikiLeaks to be funded and to carry out business with fewer limitations. However, this freedom along with enhanced interest and adoption from users means that it may aid and “facilitate money laundering, tax evasion and trade in illegal drugs and child pornography” [Three].
Bitcoin has properties which could make it significant in commerce, the most significant being low transaction costs [Five]. As there is essentially no middle man when performing transactions using Bitcoins, “there are few, if any, transaction fees associated with transfers” . This is in comparison to traditional payment methods which can have significantly higher transaction fees. Thus, in some cases, Bitcoin could provide a more feasible alternative payment method . This has implications in the developed world, for example, permitting individuals and businesses to carry out online transactions with little or no fees, reducing overall costs. In particular, for transactions which require conversions inbetween different currencies (often incurring exchange rate fees), Bitcoin could suggest a simpler and more universal payment system.
Similarly, for less economically developed countries (for plain monetary transfers inbetween two parties) services such as Western Union have traditionally been a popular way to send money back home from overseas, or to another party within the same country. A plane or percentage fee is often incurred whilst sending money. Again, Bitcoin could permit for money to be quickly and securely transferred, without the need for any extra fees. This would be hugely beneficial to those from less economically developed countries.
Traditional purchase of goods and services online is predominated by credit and debit cards, or PayPal. But where other digital currencies have failed to get a foothold, Bitcoin may not necessarily succeed. [Three] suggests that even if card use is becoming less popular, companies may be able to reduce transaction fees in general, to rival with Bitcoin. On the other forearm, Bitcoin may instead be able to establish itself as a standard in micropayments. The relative cost of processing lower value transactions is much greater for traditional payment methods, thus Bitcoin has a competitive advantage [Trio].
Bitcoin as an international payment standard has its benefits, but its volatile price suggests that it may still suffer from problems of traditional currencies. Therefore, Bitcoin could be considered as a currency exchange rate. However, some researchers argue that Bitcoin does not fulfil the criteria for it to be considered as a true currency. [Four] claim that “Bitcoin is not a denominated fiat currency”, however it has features similar to cash, for example, irreversibility and partial anonymity. According to , the wild fluctuations in Bitcoin price cannot be explained by economic and financial theory. Factors such as interest rates and inflation do not exist, as there is no central bank overseeing the issuing of Bitcoin. Thus, Bitcoin price is “driven solely by the investors’ faith in the perpetual growth” . Further to this, [Five] indicates that the three criteria for Bitcoin to be a currency, being a unit of account; medium of exchange; store of value; are not reasonably met. International use of Bitcoin is still very limited, “indicating that few people use it widely as a medium of exchange” [Five]; Bitcoin can be traded on various exchanges usually at different prices; the daily exchange against USD shows little correlation with USD exchange rate against other major currencies.
Albeit Bitcoin can be considered to be relatively fresh, there has already been some initial analysis into the cryptocurrency, and we provide a literature review here.[Two] examine the links inbetween social signals and Bitcoin price through a social feedback cycle. Using data from Bitcoin exchanges, social media, Google search trends and the user base of Bitcoin, they found two main positive feedback loops, social and user adoption cycle. An increase in popularity of Bitcoin leads to increases in searches for Bitcoin and more social media coverage. Increases in the number of users leads to an increase in Bitcoin popularity and coverage which contributes to the effect of the social cycle. However, their results fail in explaining unexpected negative switches in Bitcoin price.  studies the relationship inbetween digital currencies, such as Bitcoin, and search queries through Google Trends and Wikipedia. Price level was shown to be significantly positively related to search terms, with the relation being bi-directional, in that searches affects prices and prices affect searches.  provide an empirical analysis of Bitcoin-Exchange Risk. They note that whilst Bitcoin has seen the greatest adoption of any cryptocurrency thus far, it has also attracted the attention of criminals. Focusing on the risk of Bitcoin users from currency exchanges, their survival analysis shows that “exchange probability of closure is inversely correlated to its trade volumes” . Supporting this analysis, there is an indication that “popular exchanges are more likely to suffer security breaches” , something which one might expect. ’s analysis looked into whether Bitcoin intra-network transaction and on-exchange trading volumes are linked, and also attempts to determine if Bitcoin can be classed as an asset or a currency. Using data from two thousand eleven to 2013, including trading data, transaction data and significant Bitcoin dates, results indicate that the interest generated from fresh users of Bitcoin impacts on the volume of Bitcoins traded at the Bitcoin exchange, but not in the overall system. The authors note that as a currency, Bitcoin would need to be a “means of trade, a vehicle to store value, or a unit of account in order to compare the value of different goods or services” . Thereby hypothesising that enhanced adoption of Bitcoin will increase overall Bitcoin network volume. However, if Bitcoin is an asset, the hypothesis is that an increase in Bitcoin adoption is positively linked to an increase in Bitcoin exchange volume. Therefore, from the results it shows up that fresh users adopt Bitcoin with “speculative investment” as an objective, rather than using it as currency to purchase goods and services.
Some of the latest research comes from [Ten], modelling and predicting the Bitcoin/USD exchange rate through the application of a non-causal autoregressive model. Using data from daily closing rates of Bitcoin/USD from February 2013-June 2013, results from the analysis display that the Bitcoin/USD rate “displays gigs of local trends, which can be modelled and interpreted as speculative bubbles” [Ten]. [Ten] suggest that these speculative bubbles may arise as a result of speculative trading of Bitcoin—further supporting ’s conclusion that fresh Bitcoin users treat it as an asset. look at the structure and evolution of the Bitcoin transaction network. The probe shows two phases in the lifetime of the Bitcoin system, originally when user adoption was low, Bitcoin was “more of an experiment than a real currency”. However, after it commenced to build up momentum, Bitcoin began to behave more like a real currency. In addition, they found that in the 2nd phase the accumulation of Bitcoins through wealth distribution converges to a stable opened up exponential distribution.
The examine of [Four] measures volatility of Bitcoin exchange rate against six major currencies. Using raw annualised data over a four year period from two thousand ten to two thousand fourteen and adjusted data, taking account of volume of transactions, they find that Bitcoin shows the highest annualised volatility of percentage switch in daily exchange rates. However, accounting for the (low) volume of Bitcoin trades, volatility of the Bitcoin exchange rate is significantly diminished, displaying a more stable exchange rate. The authors note that claims of volatility and risk in Bitcoin should be interpreted cautiously. The significance of the low trading volume of Bitcoin means that the volatility of Bitcoin will emerge greater, and any trading will have a greater effect than with traditional currency.
Using the data for the period 2010–2013,  showcase “Bitcoin investment exhibits very high volatility but also very high comes back. In addition, for holders of well diversified portfolios, high risk is compensated by low correlations with other assets. Including even a puny proportion of Bitcoins in a well-diversified portfolio may dramatically improve risk-return characteristics”.
Using a known mechanism that is sturdy in detecting bubbles,  investigated the existence of bubbles in the Bitcoin market. They detected a number of short-lived bubbles over the period 2010–2014. Three of these were yam-sized appearing in the latter part of the period 2011–2013 and lasting from sixty six to one hundred six days.
Through wavelet coherence analysis,  examines Bitcoin price formation and the main drivers of price. The examine shows that factors such as “use in trade, money supply and price level” have an influence on long term price. A general increase in price attracts people to create Bitcoins, thus profit arises from the creation of Bitcoins over time. Albeit price is determined through supply and request, it is also influenced by the interest of investors. In periods of significant growth or decline in price, good and bad news were found to thrust the price further up or down, respectively.
As there is no intermediary, there is no bid-ask spread for the Bitcoin exchange rate. The lack of bid-ask spreads, that is, the absence of transaction costs, can effect the movement of quote prices, hence form the statistical properties or comes back. There is a yam-sized literature on the effects of transaction costs and bid-ask spreads on comebacks:  find evidence to suggest that “market-observed average comes back are an enhancing function of the spread; asset comebacks to their holders, net of trading costs, increase with the spread; and, there is a clientele effect, whereby stocks with higher spreads are held by investors with longer holding periods”;  find evidence to suggest that “comes back on high-spreads stocks are higher, but less spread-sensitive, than the comebacks on low-spread stocks”;  finds evidence to suggest that serial covariances of comebacks are strongly negatively correlated with the square of quoted spreads; [Eighteen] find evidence to suggest that quoted spreads are larger when larger trades take place; and so on. This suggests that any effect on the come back of Bitcoin must be related to other factors such as news relating to the digital currency.
Two latest papers on fitting of distributions to exchange rate data (not just Bitcoin) are [Nineteen] and . [Nineteen] fitted the generalized Lambda, skew t, normal inverse Gaussian and normal distributions as well as the Johnson’s family of distributions to the data.  fitted the Student’s t, asymmetric Student’s t, hyperbolic, generalized hyperbolic, generalized Lambda, skew t, normal inverse Gaussian and normal distributions to the data.
One of the known features of Bitcoin is that it is very volatile, see, for example, [Four] and . Hence, accurate fitting of its variation is so significant. The aim of this paper is to provide a formal statistical analysis of the exchange rate of Bitcoin versus the USD using a broad range of known parametric distributions in finance. The statistical analysis introduced is the most comprehensive using parametric distributions for any kind of exchange rate data.
Other motivation for this paper are: i) the exchange rate of Bitcoin to the USD behaves very differently to the exchange rate of major currencies, see Section Two; ii) there have been studies investigating the best fitting distributions for the exchange rate of major currencies, but none so far for the exchange rate of Bitcoin; iii) risk measures like the value at risk and expected shortfall can be lightly computed from the fitted parametric distributions; iv) out of sample values can be lightly predicted from the fitted parametric distributions.
The contents of this paper are organized as goes after. Section two presents the Bitcoin data used here. Some summary features of the data are described. Section three presents fifteen of the most popular parametric distributions in finance. Several of these distributions were introduced in the last few years. Section four analyzes the exchange rate data for Bitcoin using the distributions in Section Trio. Among other things future predictions are given for the exchange rate. Ultimately, some conclusions and future work are noted in Section Five.
The data are daily Bitcoin Exchange Rate on Bitstamp (Bitcoin versus USD) from the 13th of September two thousand eleven to the 8th of May 2014. The data were obtained from the database Quandl , see https://www.quandl.com/data/BITCOIN/BITSTAMPUSD Note that there are no data before the 13th of September 2011. We have chosen to use data from the Bitstamp Bitcoin exchange instead of the Bitcoin Price Index published by CoinDesk for the following reasons. We wished to concentrate on a specific Bitcoin exchange based in Britain, and one which has a significant trading volume. Bitstamp fulfils both these criteria, being located in London and being the world’s 2nd largest Bitcoin trading volume. Bitstamp exchange commenced trading on 13th September 2011, however, CoinDesk launched its Bitcoin Price Index only on 11th September 2013. Therefore, we feel that using the Bitcoin Price Index would lead to a sample size which may be too puny and unreliable to conclude any results from. The Bitcoin Price Index represents an average of Bitcoin prices across leading global exchanges. Therefore, the index is lightly affected when a certain exchange approaches a downturn or a suspension. In addition, the Bitcoin Price Index omits Bitcoin exchange if the price is not updated for more than thirty minutes. Overall, both the Bitcoin Index Price and Bitstamp exchange price overlap and go after each other very closely, so there is little difference from choosing one over the other.
The log-returns of the exchange rate are plotted in Fig one . Some summary statistics of the log-returns are given in Table one . We see that the log-returns have mean and median almost equal to zero, are negatively skewed and have a peakedness greater than that of the normal distribution. Also shown in Table one are the summary statistics of the exchange rates to USD of some of the major currencies: Australian Dollar, Brazilian Real, Canadian Dollar, Swiss Franc, Euro, British Pound and Japanese Yen. We see that the behavior of Bitcoin is sharply different compared to these currencies: its minimum is much smaller, its very first quartile is much smaller, its median is much larger, its mean is much larger, its third quartile is much larger, its maximum is much larger, its interquartile range is much broader, its range is much broader, its skewness is much more negative, its kurtosis is much more peaked, its standard deviation is much larger, its variance is much larger and its coefficient of variation is much smaller. These results are consistent with findings in [Four] and [Five].