The Fractal Nature of Bitcoin FUNDACION UNIVERSIDAD DE LAS AMERICAS PUEBLA Escuela de Negocios y Economía Departamento de Economía THE FRACTAL NATURE OF BITCOIN: EVIDENCE FROM WAVELET POWER SPECTRA TESIS PROFESIONAL PRESENTADA POR RAFAEL DELFIN VIDAL COMO REQUISITO PARCIAL PARA OBTENER EL TÍTULO DE LICENCIADO EN ECONOMÍA Asesor Dr. Guillermo Romero Meléndez Santa Catarina Mártir, Puebla Otoño 2014 1 The Fractal Nature of Bitcoin Summary. In this study a continuous wavelet transform is performed on bitcoin’s historical returns. Despite the asset’s novelty and high volatility, evidence from the wavelet power spectra shows clear dominance of specific investment horizons during periods of high volatility. Thanks to wavelet analysis it is also possible to observe the presence of fractal dynamics in the asset’s behavior. Wavelet analysis is a method to decompose a time series into several layers of time scales, making it possible to analyze how the local variance, or wavelet power, changes both in the frequency and time domain. Although relatively new to finance and economic, wavelet analysis represents a powerful tool that can be used to study how economic phenomena operates at simultaneous time horizons, as well as aggregated processes that are the result of several agents or variables with different term objectives. Keywords: Fractal Market Hypothesis, Bitcoin, Wavelet Power Spectrum, Wolfram Mathematica, Economics and Finance, Cryptocurrencies, Wavelet Analysis 2 The Fractal Nature of Bitcoin Chapter I Introduction Bitcoin is a digital currency that relies on cryptographic technology to control its creation and distribution. Just like banknotes or coins, transactions in bitcoin can be performed directly between two individuals without the need of an intermediary. However, bitcoins are not issued by any government or other legal entity, they are produced by a large number of people running computers around the world, using software that solves mathematical problems. It’s the first example of a growing category of money known as cryptocurrency. Unlike fiat currencies, whose value is derived through regulation or law and underwritten by the state, bitcoin’s technology has currency, platform, and equity properties that make it extremely difficult to assess its intrinsic value (Weisenthal 2013). As a consequence, most of bitcoin’s value is based on a highly volatile demand—what people are willing to pay and receive for them at any given time. In April 2011, less than one year after the first transactions using bitcoins took place, a single bitcoin (currency ticker BTC) was worth about $0.80 USD. Three years later, as of October 29, 2014 one bitcoin is now worth $348, having reached a historical maximum value of $1,132 in December 2013. It is widely known the bitcoin economy has experienced a recurring volatility cycle over its short existence. As media coverage on the cryptocurrency increased, this attracted new waves of investors pushing bitcoin’s price to unprecedented highs, leading to an eventual crash of the BTC/USD exchange rate. Before reaching its $1,120 historical maximum in December 2013, bitcoin’s price rose 40-fold from around $0.80 in April 3 The Fractal Nature of Bitcoin 2011 to more than $30 by June 2011 to then fall below $2 by November 2011 before stabilizing at around $5 in early 2012. After the initial boom and bust, bitcoin’s price gradually stabilized between $4.30 and $5.48 during the first half of 2012. In the second half of 2012, BTC prices climbed from $5.15 in June to $13.59 by December 2012. This pattern repeated itself twice during 2013. From $13.50 at the start of the year, bitcoin’s value soared to $237 in May and then crashed to $68 later that same month. After the first volatility cycle in 2013, BTC prices ranged between $68 and $130 until October 2013, then by the end of November bitcoin prices reached $1,120. Finally, during the first half of 2014 the USD/BTC exchange rate has steadily decreased to around $400-500. The volatility pattern observed in BTC price behavior suggests three important features in the asset’s price behavior. First, the uncharacteristically large price changes in the USD/BTC exchange rate suggest that the frequency distribution of BTC returns does not follow a normal distribution, i.e. extreme events that deviate from the mean by five or more standard deviations have a greater probability of occurrence than that predicted by the normal distribution. Fig. 1. BTC Returns Q-Q Plot 4 The Fractal Nature of Bitcoin Figure 1 shows the Quantile-Quantile Plot for BTC historical returns and illustrates the evidence of long tails and over dispersion in the series, represented by the blue thick line. Second, clear clustering periods of high and low volatility in the BTC price data suggest that while asset returns may be random, its periods of volatility are not. This is illustrated in Figure 2. The top graphic shows the autocorrelation of BTC returns, suggesting no sign of serial correlation between returns. The bottom graphic shows the correlation of BTC volatility, i.e. the second moment of the asset’s returns. Fig. 2. Autocorrelation of BTC returns and volatility 5 The Fractal Nature of Bitcoin The second graphic in Figure 2 shows a clear positive trend in the autocorrelation of the asset’s volatility, a clear sign of long memory, or persistent behavior. Finally, bitcoin price data exhibits evidence of scale invariance, or self-similar statistical structures, at different price levels. For example, BTC returns follow the same frequency distribution regardless of time scale; while bitcoin’s price volatility cycles show the same behavior, independent of price level. These features directly violate the fundamental assumptions of Gaussian distribution required by the established Efficient Markets Hypothesis (EMH), rendering most financial modelling approaches unsuitable to study bitcoin price behavior. Moreover, after decades of statistical analysis of price fluctuations across different markets, asset types, and time periods, there is a large number of studies documenting the failure of EMH to mirror or model the empirical evidence of financial time series (Mandelbrot 1963, 1997; Blackledge 2010). Despite the widespread use of the Brownian motion and Gaussian distribution paradigms in financial economics, a number of systematic statistical departures from the EMF have been identified and are now widely acknowledged as “stylized facts” of financial time series (Rama 2001, 2005; Borland et al. 2005; Ehrentreich 2008; Dermietzel 2008). Notably, the main stylized facts standing out in the literature include the three prominent features of bitcoin’s volatility cycle previously mentioned: heavy tails or nonnormal distribution of returns, long memory effect in squared returns also known as volatility clustering, and presence of fractal dynamics. Therefore, given the strong deviations from the EMH framework readily observable in the BTC price data, an alternative analytical framework is used to study financial data with likely presence of non-normality, self-similarity, and persistent volatility. 6 The Fractal Nature of Bitcoin The Fractal Market Hypothesis (FMH) is a theoretical framework developed by Peters (1991a, 1991b, 1994) where he proposes a more realistic market structure that places no statistical requirements on the process; explain why self-similar statistical structures exist; and how risk is shared and distributed among investors (1994, 39). Under the FMH approach market stability is maintained only when many investors participate and they can cover a large number of investment horizons, thus ensuring ample liquidity for trading (44). Peters argues that after adjusting for scale of investment horizon, all investors must share same risk levels, which explains why the frequency of distribution of BTC returns exhibits self-similar behavior at different scales (46). According to the FMH, a market becomes unstable when its self-similar structure breaks down, i.e. when investors with long term horizons either stop participating in the market or become shortterm investors themselves. When long-term fundamental information is no longer important or unreliable, markets become unstable and are characterized by extreme high levels of short-term volatility. This approach explains the presence of periods of clustering volatility in the BTC time series and the occurrence of extreme events that violate the normality of the frequency distribution of bitcoin returns. The FMH suggests that stable markets are characterized by equally representation of all investment horizons in the market so that supply and demand are efficiently cleared. When investors at one horizon (or group of horizons) become dominant, the selling or buying signals of the investors at these horizons will not be met with a corresponding order of the remaining horizons and periods of high volatility might occur. Thanks to time-frequency analysis it is possible to investigate whether BTC returns follow the market dynamics established by the fractal markets hypothesis and its focus on liquidity and investment horizons. According to Kristoufek (2013), after performing a continuous 7 The Fractal Nature of Bitcoin wavelet transform to a time series and obtaining its wavelet power spectra it is possible detect the dominance of specific investment horizons during periods of high volatility. Wavelet analysis is a method to decompose a time series into time-frequency space, it uses mathematical expansions that transform data from the time domain into different layers of frequency levels. This makes possible to observe and analyze data at different scales. Although this approach is relatively new to economics, wavelets have been used in a wide range of fields. For example, for the analysis of oceanic and atmospheric flow phenomena in geophysics (Torrence and Compo 1998), image processing for computer and image compression (Grapps 1995), as well as in medicine for heart rate monitoring (Thurner, Feurstein, and Teich 1998), and for molecular dynamics simulation and energy transfer in physics (McCowan 2007) just to name a few. Among the most well-known applications of wavelet analysis are the FBI algorithm for fingerprint data compression and the JPEG algorithm for image compression (Grapps 1995, Li 2003). Scope of this thesis The main goal of this study is to provide empirical evidence supporting the Fractal Market Hypothesis. To do so, the BTC returns time series is analyzed to determine the existence of dominance of short investment horizons during periods of high market turbulence. This objective is accomplished using a continuous wavelet transform analysis to obtain information about bitcoin’s price volatility across time and different scales of investment horizons. There are several reasons for the importance of this study. First, to date this is the only study using wavelet analysis to detect dominance of investment horizons in BTC price returns. Second, the results of the continuous wavelet transform of the time series show supporting evidence in favor of the Fractal Market Hypothesis. Third, the wavelet 8 The Fractal Nature of Bitcoin analysis performed suggests that while bitcoin’s price has been characterized by high volatility, it follow the same market dynamics as other currencies and equity markets (e.g., government bonds, stocks, and commodities). Finally, the use of wavelet to analyze economic phenomena is relatively recent, this work will show original contributions to the applications of wavelet analysis in economics, finance and cryptocurrencies.1 Organization of the manuscript The remainder of this work is organized as follows. The Theoretical Framework section presents an overview of the Bitcoin payment system and an introduction Wavelet Analysis and its relation to FMH. The following section presents the continuous wavelet transform methodology, results and discussion. Then, the final section concludes with a general discussion, future research subjects, and benefits of wavelet analysis to the study of economic phenomena. 1 This manuscript is based on the undergraduate thesis project of the first author (Delfin 2014) and supervised by the second author. 9 The Fractal Nature of Bitcoin Chapter II Theoretical Framework This background section should provide an introductory understanding to the topics presented in the following sections, although it is far from a complete examination of the concepts covered in this study. Although the intersection of these subjects has yet to gain wider recognition, studies on Bitcoin’s public ledger technology along with Wavelet Analysis span several fields within economics and finance in general. It is highly recommended to consult the sources referenced in this section should the reader be interested in a more comprehensive understanding of the topics covered in this work. 2.1 The Bitcoin Protocol Bitcoin is a peer-to-peer payment system introduced as open source software in January 2009 by a computer programmer using the pseudonym Satoshi Nakamoto (Nakamoto 2009). It is referred a cryptocurrency because it relies on cryptographic principles to validate transaction in the system and ultimately, control the production of the currency itself. Each transaction in the system is recorded in a public ledger, also known as the Bitcoin block chain, using the network’s own unit of account, also called bitcoin. 2 The block chain ledger is a database where transactions are sequentially stored and the file containing it is visible to all members on the network. 2 According to the Bitcoin wiki website (https://en.bitcoin.it/wiki/Introduction#Capitalization_.2F_Nomenclature), capitalization and nomenclature can be confusing since Bitcoin is both a currency and a protocol. Bitcoin, singular with an upper case letter B, will be used to label the protocol, software, and community, and bitcoins, with a lower case b, will be used to label units of the currency. 10 The Fractal Nature of Bitcoin Bitcoin’s block chain is a unique technology since it solves several problems at once: it avoids forgery or counterfeiting, it also avoids the need for a trusted intermediary, and regulates the creation of new bitcoins in a controlled way (Congressional Research Service 2013; Velde 2013). Since validation for each transaction is a computationally intensive task, the Bitcoin protocol solves these problems by rewarding those who devote computing power to validate transactions with the privilege to create new bitcoins in a controlled way. According to Barber et al. (2012), there are several reasons why Bitcoin, despite more than three decades of previous attempts at digital money by cryptography researchers (see for example Chaum 1983; Chaum, Fiat, and Naor 1990; Szabo 2008), has witnessed enormous success since its invention. Among the number of reasons are: no central point of trust, economic incentives to participate, predictable money supply, divisibility and fungibility, transaction irreversibility, low transaction costs, and readily available implementation. Contrary to earlier implementations of e-cash, Bitcoin is a decentralized network that lacks a central trusted entity. The network assumes that the majority of its nodes are honest, and as mentioned earlier, the task of validating transactions for dispute resolution and to avoid double spending are carried out by members on the network dedicating computing power for those purposes. The absence of a central point of trust guarantees that the currency cannot be subverted by any single entity—government, bank or authority—for its own benefit, and while this feature can be used for illegal purposes there are also numerous legitimate reasons for using this technology. Regarding the economic incentives for participation in the Bitcoin network, Kroll, Davey, and Felten (2013) argue if all parties act according to their incentives the Bitcoin protocol can be stable, meaning the system will continue to operate. Since the generation 11 The Fractal Nature of Bitcoin of new bitcoins is rewarded only to those individuals who devote computing power to validate transactions, also known as bitcoin mining, this reward ensures that users have clear economic incentives to invest unused computing power in the network. In addition to rewards from dedicating computational cycles to verify transactions, miners can charge small transaction fees for performing said validation. Finally, Barber et al. (2012, 3) argue the open-source nature of the project also gives incentives for new applications within the protocol and the creation of a large ecosystem of new businesses. For example, new applications that add better anonymity measures or payment processing services that allow merchants to receive payments in bitcoin, send money internationally at significant low cost. In addition to a predictable money supply, Barber et al. (2012) argue that the divisibility, fungibility, and transaction irreversibility of Bitcoin gives it an advantage over other e-cash systems since the coins can be easily divided, up to eight decimal places, and recombined allows to create a large number of denominations; while the irreversibility of transactions means that merchants concerned with credit-card fraud and chargebacks can conduct business with customers in countries with high prevalence of credit card fraud. Moreover, thanks to its high divisibility Bitcoin has great potential as a platform for enabling micropayments, payments much smaller than what the traditional financial system can handle. After Nakamoto’s publication of the Bitcoin protocol in January 2009, the homonymous currency remained a modest project undertaken by a small community of cryptographers during its first year. However, Nakamoto’s creation soon spreaded beyond the initial community and took a life of his own. In October 2009 the first USD/BTC exchange rates were published by New Liberty Standard (2009), $1 was valued at 1,309.03 BTC. In May 2010, Laszlo Hanyecz, a Florida programmer, conducted 12 The Fractal Nature of Bitcoin what is thought to be the first real-world bitcoin transaction, agreeing to pay 10,000 bitcoins for two pizzas from Papa John’s worth around $25 at the time (Mack 2013). Two months later in July 2010, bitcoin’s exchange value began a 10x increase over a 5 day period, from about $0.008/BTC to $0.08/BTC. By November of that year bitcoin had reached a market capitalization of $1 million while the exchange rate was $0.50 for 1 bitcoin (Bitcoin 2014). The next important milestone for the currency occurred in February 2011 when bitcoin reached parity with the US dollar at the now defunct Japanese exchange MtGox. During the spring of 2011 after several stories on the new cryptocurrency by high profile media outlets, one from Time (Brito 2011) another one by Forbes reporter Timothy Lee (2011), and also from popular design and technology blog Gizmodo (Biddle 2011), the price of bitcoin skyrocketed from around 86 cents in early April to $9 at the end of May. Additionally, in June 1st media outlet Gawker published a story about the use of bitcoin in the online black market Silk Road to buy drugs, weapons, and stolen personal information thanks to the currency’s pseudoanonymous features (Chen 2011a, 2011b). One week later bitcoin’s exchange rate increased three-fold from $9/BTC to $31/BTC. As the price of bitcoin rose and stories of return on investment in the order of thousands, mining became more popular. Now real money stakes and the dramatic price rise had attracted people who saw bitcoin as a commodity in which to speculate. However, given the novelty of this asset and how its uncharacteristic behavior clearly violates the fundamental assumptions of most financial modelling approaches, an alternative analytical framework is used to study bitcoin price behavior. 13 The Fractal Nature of Bitcoin 2.2 Wavelet Analysis and the Fractal Market Hypothesis As mentioned in the introductory section, the FMH suggests that stable markets are characterized by equally representation of all investment horizons while market volatility occurs when the selling or buying signals of a dominant investment horizon are no met with a corresponding order from the remaining horizons. However, simultaneous operation at different time horizons it’s not only restricted to currency and equity markets. Aguiar-Conraria and Soares (2011, 1) argue that many economic processes are the result of actions of several agents who have different time objectives and therefore, many economic time series are an aggregation of components operating on different frequencies (spanning milliseconds in High Frequency Trading to several decades for institutional investors). Moreover, Ramsey and Lampart (1997a) argue that economists have long acknowledged the importance of time scale but only until recently it had been difficult to decompose economic time series into time scale components. Central banks for example have different objectives in the short and long run, and therefore operate at different time scales. The main advantage of using the continuous wavelet transform (referred as CWT from now on) in economic time series is its ability to analyze how the wavelet power of the underlying process changes in both the time and frequency domain. In terms of financial economics, the wavelet power spectrum (WPS) is defined by Rua (2012) as the contribution to the variance around each time and scale. Formally the WPS is defined as the squared absolute value of the wavelet coefficients resulting from the transform. According to FMH a high power spectrum is associated with dominant investment horizons, i.e. the selling or buying signals of investors at the dominant horizons are not being met with a corresponding order from the remaining horizons and periods of short- 14 The Fractal Nature of Bitcoin term volatility might occur. Therefore, high power spectrum values should be observed at low time scales (high frequencies) during periods of high volatility. 2.3 Origins of Wavelet Analysis In order to talk about wavelet analysis it is necessary to talk about Fourier analysis first since the former has various points of similarity and contrast with the later. The Fourier transform is based on using a sum of sine and cosine functions of different wavelengths to represent any other function. The Fourier transform of a time series is a function ℱ in the frequency domain ℱ angular frequency and −� = = ∫−∞ ∞ −� , where is the according to Euler’s formula. − However, the Fourier transform does not allow the frequency content of the signal to change over time, making it unsuitable for analyzing processes that have time-varying features. This means that if a single frequency is present in a process but it varies over time the Fourier transform does not allow to identify when in time the frequency component changes (Rua 2012). To illustrate the shortcomings of the Fourier transform when reproducing signals that have time-varying features, the following example is based on Wolfram’s presentation on wavelet concepts (see Wolfram 2014a). Considering the stationary process = [ �] + [ �] . This process is composed of two signals, one at 20 Hz and another at 40 Hz. When the Fourier transform of this data is performed two frequencies are correctly identified, at two times the frequency in the x-axis, i.e. 40 and 80 Hz respectively (see Figure 3). While the Fourier transform provides frequency information it lacks time information about these frequencies, i.e., at what time did these frequencies occur and for how long? Considering now a non-stationary process with three frequency components defined by ′ ={ [ �] [ �] [ �] ≤ ≤ ≤ ≤ ≤ ≤ , the Fourier transform 15 The Fractal Nature of Bitcoin correctly shows three peaks (Figure 3) at the corresponding frequencies (1, 5, and 10 Hz), however, the transform does not provide information about the time varying components of this process. According to McCowan (2007, 6), the Fourier transform gives optimal results only when a single frequency is present. When multiple frequencies are present in a process, the transform may have difficulties separating noise or assigning accurate relative amplitudes for each frequency. A possible way to overcome the previous limitations is the short-time or windowed Fourier transform, a Fourier-related transform used to obtain frequency information of local sections of a signal as it changes over time. As its name suggests the Fourier transform is performed for short periods of time, sliding a segment of length across all the data. However, the windowed Fourier transform (WFT) imposes the use of constant-length windows. Fig. 3. Stationary and Non-Stationary processes and their Fourier Transform 16 The Fractal Nature of Bitcoin This restriction makes the WFT an inaccurate method for time-frequency analysis since many high and low-frequency components of the process or signal will not fall within the frequency range of the window. Relatively small windows will fail to detect frequencies whose wavelengths are larger than the size of the window while relatively large windows will decrease the temporal resolution because larger intervals of signal are analyzed at once. Torrence and Compo (1998, 63) argue that for analyses where a predetermined scaling may not be appropriate because of a wide range of dominant frequencies are present in the process, a method of time–frequency localization that is scale independent, such as wavelet analysis, should be employed. 2.4 The Continuous Wavelet Transform Just as the windowed Fourier transform, the aim of the continuous wavelet transform (CWT) is to detect the frequency, or spectral, content of a signal and describe how it changes over time. The CWT however uses a base function that can be stretched and translated with a flexible resolution in both frequency and time, making it possible to analyze non-stationary time series that contain many different frequencies. Moreover, the CWT intrinsically adjusts the time resolution to the frequency content. This means the analyzing window width with will narrow when focusing on high frequencies (short time periods) and widen when assessing low frequencies (long time scales). The CWT of a discrete sequence +∞ ∗ ∫−∞ wavelet �, �, can be formally defined as , the CWT decomposes the time series ∗ �, = , where * denotes the complex conjugate. Starting with a mother in terms of analyzing wavelets . The analyzing wavelets are obtained by scaling and translating defined as �, = √| | −� , which is , where s is the scale and � the translation parameters. 17 The Fractal Nature of Bitcoin The wavelets can be stretched (if | | > ) or compressed (if | | < ), while translating the wavelet means shifting their position in time. Thanks to the CWT flexible resolution in both frequency and time, rapidly changing feature can be capture at low scales, or wavelengths, whereas slow changing, or higher time scales, components can be detected with dilated analyzing wavelets (Torrence and Compo 1998; Aguiar-Conraria and Soares 2011). Mother wavelets must fulfill certain mathematical criteria in order to be considered analytical wavelets, in economics and finance the Morlet wavelet is the most widely used mother wavelet (Torrence and Compo 1998; Aguiar-Conraria and Soares 2011; Rua 2012; Kristoufek 2013). The Morlet wavelet consist of a complex sine wave modulated by a Gaussian envelope, and its formally defined as: The term = � −4 ℯ � � ℯ− . controls the nondimensional frequency, i.e. the number of oscillations within the Gaussian envelope, and is set equal to six to satisfy the admissibility criteria as analytic wavelet (see Lee and Yamamoto 1994; and Adisson 2002; for a detailed analysis of wavelet admissibility criteria). Figure 4 shows the Morlet Wavelet, which unlike sines and cosines, it is localized in both time and frequency. Real Part Imaginary Part Fig. 4. Morlet Wavelet 18 The Fractal Nature of Bitcoin 2.5 Wavelet Power Spectrum and other Definitions in the Wavelet Domain Once the CWT has been defined, we offer two definitions from the wavelet domain to analyze an asset’s volatility as well as its local covariance with other assets (see Ranta 2010 for additional definitions regarding correlation and contagion in the timefrequency domain). First, the wavelet power spectrum (WPS) can be defined as | �, | , i.e. the square of the absolute value of each coefficient at each time and scale, and measures the local contribution to the variance of the series. Second, the crosswavelet transform (XWT) of two time series transforms �, and �, and , is defined as corresponding cross-wavelet spectrum is defined as , with continuous wavelet �, � = =| �, ∗ �, . The | . According to Aguiar-Conraria and Soares (2011, 16), the cross-wavelet power of two time series can be defined as the local covariance between them in the time-frequency domain, giving the researcher a quantified indication of the similarity of volatility between the time series. 19 The Fractal Nature of Bitcoin Chapter III Methods and Results In this section the CWT will be implemented on the BTC historical returns time series to provide evidence for the dominance of short investment horizons during periods of high volatility. Additionally, since all the analysis in this study was performed using the computational software Mathematica, the code used to perform the computations will be used to provide the reader new tools for wavelet analysis. The findings of this analysis will be discussed afterward. 3.1 Data A time series for the price of bitcoin against the U.S. Dollar will be analyzed to find their respective wavelet power spectrum. The oldest available date for bitcoin prices is July 17, 2010. The time series cover the oldest available BTC price until October 29, 2014. Additionally BTC prices are compared against Litecoin (LTC) prices, both in USD. Just as Bitcoin, Litecoin is a peer-to-peer cryptocurrency inspired by Bitcoin and introduced as an improvement on it. As of November, 2014 Litecoin has the third largest cryptocurrency market capitalization with approximately $128,786,179. Bitcoin is the leading cryptocurrency in terms of market capitalization ($5,090,366,912), followed by the Ripple protocol ($157,341,776) and Litecoin. Cross-wavelet transformation will be used to study the local covariance in the time-frequency domain between the BTC and LTC returns time series. Data was not sufficiently available to perform wavelet analysis on the Ripple (XRP) currency. 20 The Fractal Nature of Bitcoin The data used for this study was obtained from the data platform Qandl’s website, a search engine for numerical data with access to a large collection of financial, economic, and social datasets. 3.2 Method: Basic Wavelet Concepts Performing a CWT in Mathematica can be done with very few commands. Before the main analysis of this study, three examples will be presented to overview basic wavelet transform concepts and their advantage over time or frequency analysis. The first example is based on Aguiar-Conraria and Soares (2011, 4). 50 years of monthly data are generated according to the process: = and = � = ≤ ≤ for + � , and = +� ; = , ,…, otherwise. It can be seen this process is the sum of two periodic components: a ten-year cycle and three-year cycle that briefly changes to a five-year cycle during between the second and third decades. Although Figure 5 shows the process in the time domain, it is not possible to clearly observe any the cyclic dynamics of the series. Figure 5. Time series � � Figure 6 shows a visualization, also called wavelet scalogram, of the wavelet power spectrum, | �, | , of the process. The wavelet scalogram functionality in 21 The Fractal Nature of Bitcoin Mathematica plots the absolute value of the wavelet transform coefficients at each time and scale. In Figure 6 the wavelet scalogram is able to capture the three cyclic dynamics of the time series. The time dimension is represented in the horizontal axis while the vertical axis represents the scales, or frequencies, analyzed. The wavelet power is represented by color, ranging from blue for low power to red for high wavelet power. The lower region in red from Figure 6 shows the ten-year cycle of the time series, while the light green regions in the middle section of the graphic shows how the second component of transitions from a three-year cycle to a five-year cycle between the second and third decade. Since the series is given in monthly data, the second and third decades fall within observations 240-360 on the horizontal axis. Fig. 6. Wavelet Power Spectrum of The following two examples are based Wolfram’s presentations on wavelet concepts and applications (2014a, 2014b). The next example of this section focuses on a CWT of a non-stationary process composed of multiple frequencies. Four different frequencies will be operating at different instances in time. The series will be generated by the process 22 The Fractal Nature of Bitcoin ≤ ≤ [ �] ≤ ≤ [ �] = The plot of process [ �] { [ �] ≤ ≤ ≤ ≤ is shown in Figure 7, the four distinct frequencies can be clearly observed as t increases. Figure 8 shows each frequency composing the process operating at different frequency bands, as t advances the bands move up the wavelet scalogram, indicating the time series is operating at increasing frequencies. Fig. 7. Time Series with Four Frequency Components Fig. 8. Wavelet Power Spectrum of The third and final example of this section is used to illustrate how discontinuities, or in economic terms structural changes and regime shifts, can be identified using wavelet analysis. Considering a process ℎ determined by a simple cosine function. 23 The Fractal Nature of Bitcoin Overlapping the cosine function an extreme event of small duration occurs at time t, hence the process is defined as: ℎ CWT on the series ℎ = [ �] + ℯ^[− 5 ∗ − ]. After performing a the wavelet scalogram can provide a clear picture of the process’ behavior in the time-frequency domain. Figure 9 shows the wavelet power of the series at various frequencies or scales. The left indexes on the vertical are associated with each scale while the right indexes represent the voice per scale. At large scales (low frequencies), the wavelet scalogram is able to capture the signal from the cosine function but as we move upward to lower scales (higher frequencies), the extreme and short-lived event can be localized in both frequency and time. Fig. 9. WPS of a Cosine Signal Overlapped by an Extreme and Short-Lived Anomaly 3.3 Wavelet Transform of Bitcoin’s Returns Once the basic concepts of wavelets analysis in the Mathematica platform have been established, the BTC returns time series will be decomposed using the CWT. The first steps are importing bitcoin historical prices to Mathematica, defining the time series for bitcoin returns, and creating a list with the data. The following three lines of code perform each step respectively: btcprice=Import["C:\\Users\\...\\BTCAVERAGEUSD.xlsx",{"Data",1,{All},2}]; returns[x_]:=Log[(btcprice [[x+1]]/ btcprice [[x]])]; btcreturns=Array[returns,(Length[btcprice])-1]; 24 The Fractal Nature of Bitcoin After the BTC returns data is defined, a CWT can be applied with the following command: cwt=ContinuousWaveletTransform[btcreturns,MorletWavelet[], {9,10}]. The CWT command gives the wavelet transform of btcreturns, using the complex MorletWavelet[], and decomposes the data into nine octaves, or scales, and ten subsequent voices, or samples for each scale. The scales chosen for the wavelet = transform are defined as fractional powers of two: ∗� , where /� � ; = , , … , �; and � = is the smallest resolvable scale and � determines the total number of layers in which the signal will be decomposed, i.e. � = # � � Additionally, the smallest resolvable scale ∗ #� . is computed automatically as the inverse of Fourier wavelet length of the wavelet (Wolfram 2014c). For the CWT of the btcreturns time series the smallest resolvable scale computed is .86 days or about 20 hours, therefore the scales and samples per scale will be computed as = , ,…, . =. ∗ /9 , By default Mathematica computes the number of scales used in each transform as � � , where N is the length of the time series, while the default value for the number � of voices per octave is four. Computing the � , where � = for the BTC returns time series results in 9.59. Mathematica correctly computes the number of scales to be used in the wavelet transform, however the number of scales was explicitly indicated in the CWT command in order to specify the number of voices per scale as well. The more voices per scale are used in the CWT the better the time-frequency resolution, hence it was increased to ten from the default value of four. Mathematica evaluates the CWT command and the output Continuous Wavelet Data object (CWDo) in the form {{ ,� }→ ,…, �} , with N wavelet 25 The Fractal Nature of Bitcoin coefficients corresponding to { ,� } . The CWDo also contain additional information that can be later accessed and manipulated. For example, each octave and voice pair is associated with a certain scale, these can be accessed using the property “Scales”: cwt["Scales"] {{1,1}0.925992, {1,2}0.992453, …, {5,1}14.8159, {5,2}15.8793, …, {9,9}412.735, {9,10}442.358} 3.4 Wavelet Power Spectrum of Bitcoin’s Returns As mentioned in previous sections, the scalogram is a visual method to represent the absolute value of each coefficient, or wavelet power. The wavelet scalogram displays three axes: the horizontal axis represents time, the vertical axis the time scales or frequencies, and the transform’s coefficient values. The coefficient values are plotted as rows of colorized rectangles whose color corresponds to the magnitude of each coefficient. Figure 10 shows three graphics. The middle graphic shows the wavelet scalogram for the bitcoin returns CWT. The top graphic shows the historical BTC returns and the bottom graphic is a plot of the historical observed volatility. The regions with significant wavelet powers against the null hypothesis of a white noise (AR[1] process) are denoted by orange and yellow colors. According to Torrence and Compo (1998); and Aguiar-Conraria and Soares (2011), the use of CWT for finitelength series will suffer from border distortions at the beginning and end of the wavelet power spectrum because the wavelet function will be defined beyond the limits of the time series. The cone of influence (COI) is the region in the time-frequency plane where border distortions become important, and in Figure 10 by the region above the white 26 The Fractal Nature of Bitcoin contour line. The COI can be defined as the set of all observations t included in the effective support of the wavelet at a given position and scale. This set is defined by | − �| ≤ √| | −� where � is the translation parameter of the analyzing wavelets ∗ �, = , s is the scale parameter and [− , ] is the effective support of the daughter wavelets, i.e. the initial and final values of the time series [1 , 1548]. Fig. 10. Wavelet Power Spectrum of Bitcoin Returns 27 The Fractal Nature of Bitcoin As mentioned at the beginning of this section, the scalogram displays the wavelet transform in three dimensions: time, frequency, and wavelet power. Figure 11 shows a three dimensional representation of the BTC returns power spectrum. Fig. 11. Three-Dimensional Representation of Bitcoin’s WPS Several features can be observed in the previous two figures. First, the highest wavelet power regions (colored in red, orange, and yellow) are associated with periods of highest volatility. This can be confirmed with the top and bottom graphics of Figure 10, where returns and volatility are respectively plotted. Second, for most of the analyzed period no investment horizon, or scale power, dominates the series. However, the wavelet scalogram correctly captures the biggest price movements in bitcoin: the 40-fold increase around mid-2011 from around $0.80 to more than $30, a low variance period during 2012, and the two price bubbles from 2013 during May and late November. Third, during the periods of high volatility the BTC power spectrum show clear dominance of short investment horizons. Moreover, these dominant investment horizons are located within 28 The Fractal Nature of Bitcoin the 3.5 to 7 days band, and during the price increase in May 2013 dominant investment horizons can also be observed in the 7 to 14 days band. Larger investment horizons (lower frequencies) only show moderate wavelet power. However, since the cryptocurrency was created little less than six years ago it is not possible to draw significant conclusions for large investment horizons. Finally, the results presented in Figure 10 support the Peter’s (1991a, 1991b, 1994) thesis of dominant investment horizons during periods of turbulence and provide further evidence in favor of FMH’s prediction of market stability only under equally representation of all investment horizons. According to the FMH, a market becomes unstable when its self-similar structure breaks down. This can happen for a number of reasons, if investors with long term horizons stop participating in the market, become short-term investors themselves, or when long-term fundamental information is no longer important or unreliable. Given the novel nature of Bitcoin, the large price swings and low liquidity of cryptocurrencies in general might make clearer the fractal dynamics of these markets. Indeed, after closer examination of the wavelet scalogram it is possible to magnify certain regions of the time-frequency plane to observe the presence of fractal dynamics in the series. The following section will present evidence of bitcoin’s self-similar behavior in the time-frequency plane. 3.5 Self-Similarity in Bitcoin Returns Contrary to their mathematical counterparts, real life fractal processes exhibit selfsimilar behavior over a finite range of scales. Bitcoin returns time series however exhibit fractal properties over a sufficiently large range of scales to allow wavelet transform analysis to examine the process. Since a process with fractal behavior displays selfsimilar structures regardless of scale, wavelet analysis is adequately suited to detect these 29 The Fractal Nature of Bitcoin properties. The basic principle for studying fractal processes with wavelet transform is that since the signal is self-similar at any scale, the wavelet coefficients of the transform too will be self-similar and this can be observed plotting the power spectrum of the signal or series. In order to show the self-similarity of BTC returns in the time frequency domain, first only the real values of each wavelet coefficient are taken: Rcwt= ReplacePart[cwt,1Re[cwt[[1]]]]; Once the real part of the wavelet transform is defined the wavelet scalogram is plotted and shown in Figure 12 Self-similar curves in the time-frequency plane are visible at first glance. The fractal pattern is present throughout the series, irrespective of scale or wavelet power. Fig. 12. Self-Similar Behavior of Bitcoin Price Returns Figures 13a to 13d are magnifications of Figure 12 at varying scales. The top left chart depicts scales 1 to 4, the top left figure scales 4 to 7, while the bottom left chart 30 The Fractal Nature of Bitcoin shows scales 5 to 8, and the bottom right chart scales 7 to 10. Specific scales can be plotted as follows: c1=WaveletScalogram[Rcwt,{1|2|3|4,_}]; c2=WaveletScalogram[Rcwt,{4|5|6|7,_}]; c3=WaveletScalogram[Rcwt,{5|6|7|8,_}]; c4=WaveletScalogram[Rcwt,{7|8|9|10,_}]; Grid[{{c1,c2},{c3,c4}}] Fig. 13. Self-Similar Behavior of Bitcoin Price Returns at Varying Scales 31 The Fractal Nature of Bitcoin Chapter IV Conclusion Concluding remarks are presented in this section. The contribution of wavelets analysis to the Fractal Market Hypothesis and the economic sciences in general are discussed, as well as the future possible areas of research using time-frequency analysis. 4.1 Fractal Market Hypothesis: Evidence from Wavelet Power Spectrum In spite of the novelty of the Bitcoin protocol and the uncharacteristically high volatility of the homonymous currency, the predictions made by the Fractal Market Hypothesis correctly capture the asset’s behavior. Thanks to the ability of wavelet analysis to decompose a time series into different scales it is possible to observe the dominance of short investment horizons during periods of price volatility. The theoretical framework developed by Peters (1991a, 1991b, 1994) takes into account heterogeneous agents who operate at simultaneous time horizons and react to market information with respect to their investment horizon, therefore it is possible to account for the statistical departures to the Efficient Market Hypothesis observed in the cryptocurrency’s returns. Additionally, the use wavelet analysis allowed to observe the presence of selfsimilar dynamics in the time series through the wavelet power spectrum. This methodology has also been used to detect fractal properties in a wide range natural phenomena, from fluid turbulence, to DNA sequences, and breathing rate variability (Addison 2002). 4.2 Wavelet Analysis in Economics and Finance Many authors argue the importance of multiscale relationships in economics and finance (In and Kim 2012; Aguiar-Conraria and Soares 2011; and Ramsey and Lampart 32 The Fractal Nature of Bitcoin 1997). In financial terms, market dynamics are affected by investment horizons that range from high frequency trading to individual stock brokers, hedge funds, multinational corporations, pension funds, and government debt. However, despite the wide range of investment horizons operating in the market, most economic analysis has relied on only two scales, short and long run. The use of time-frequency analysis is being rapidly adopted in economics to study how a process operates across a wide range of time scales. Wavelet analysis has been used to investigate the multiscale relationship between the stock and futures markets over various time horizons, the interest rate swap market in the time-frequency domain, long memory in rates and volatility of LIBOR, and the relationship between stock returns and risk factors at various time scales (In and Kim 2012) Ramsey (2002) lists many benefits of incorporating wavelet analysis to the discipline, for example estimators for novel situations, greater estimation efficiency, robustness of modelling error, reduction of biased estimations, and most importantly, discovering new insights into the properties of economic phenomena. For example, Ramsey (2002) mentions previous studies using wavelet analysis, or time-scale decomposition, to study the term structure of interest rates (Ramsey and Lampart 1997b), the distinction between permanent and transitory shocks, or the relationship by time-scale of money and income, and expenditure and income (Ramsey and Lampart 1998a, 1998b). In their studies of money, income, and expenditure in the time-frequency domain Ramsey and Lampart (1998a, 1998b) found evidence of complex behavior in the relationships between these variables. The authors found that the delays observed between variables are a function of time and scale, contrary to the accepted assumption that delays between variables are fixed. This provides opportunities for future research examining the underlying mechanisms of the time-varying delays. Ramsey (2002, 16) 33 The Fractal Nature of Bitcoin speculates the “timing” of actions by economics agents can explain time-varying delays and provides as an example a 2001 push by auto-manufacturers to lower the purchase price on cars. The author argues the automakers’ decision had two effects. First, undoubtedly increased quantity demanded in reaction to an implicit price decline, but it also shortened the delay between income and expenditure. Wavelet analysis can also be used to study structural change and regime shifts, for example, to model the impact of minimum wage and tax legislation, and innovation. Aggregate time series can be decomposed into long term structural components, medium term seasonal components, and short term random components. This approach can allow to characterize a robust system at high scales that also permits fluctuations that are not entirely random and shorter time scales. As it was shown in Figure 6, wavelet analysis allows for the study of transitional changes that were previously impossible to observe thanks to wavelet’s ability to capture hidden dynamics. Ramsey (2002, 22) also points to the analysis of term structure of interest rates as a field where wavelet applications should provide extensive and deep insights since the role of the horizon of the decision maker on market outcomes is so clearly indicated. Kiermeier (2014) for example, analyses the risk factors of the European term structure of interest rates and find good forecasting results, with up to one month of significant forecasts even during times of financial market distress. Time-frequency analysis also allows for new developments on forecasting. By decomposing a time series into its global and local aspects, specific forecasting techniques can be applied to each scale of the time series. 4.3 Future Research The results presented in this study indicate new areas for both empirical and theoretical research. For example how agents operate at several scales simultaneously, 34 The Fractal Nature of Bitcoin both at the individual and aggregate level; or what are the long and short term structural components underlying cryptocurrencies and their relationship to other economic variables in the time-frequency domain. The application of wavelet analysis to economics and finance is still in its infancy when compared with other fields. However, the wavelet literature in economics is rapidly growing and expanding to other areas in the fields. 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