Modeling the price of Bitcoin with fractional Brownian motion: a Monte Carlo approach
@PhoenixGruppeMariusz Tarnopolski Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Krak ´ow, Poland
Abstract The long-term dependence of Bitcoin (BTC), manifesting itself through a Hurst exponent H > 0.5, is exploited in order to predict future BTC/USD price. A Monte Carlo simulation with 105 fractional Brownian motion realisations is performed as extensions on historical data. The accuracy of statistical inferences is 20%. The most probable Bitcoin price in 180 days is 4537 USD.
1. Introduction
Bitcoin, after being introduced by Nakamoto (2009), has been gaining popularity. Its price and volume have increased by several orders of magnitude. Being the so-called cryptocurrency, i.e. an asset derived from mathematical cryptography, it is based on a new technology called the blockchain (Bradbury, 2013; Ali et al., 2014). Its other fundamental characteristics are: being decentralised, and having a fixed total number of coins: 21 million, with more than 16 million already in circulation. Bitcoin is still very young, especially considering the fact that the last coin is to be mined around year 2140 (Vranken, 2017). While Bitcoin does not have the features of a currency (Baek & Elbeck, 2015; Dyhrberg, 2016) and is highly volatile (Dwyer, 2015; Katsiampa, 2017), it can serve as an effective diversifier (Briere et al., 2015; Bouri et al., 2017). Moreover, it recently ` entered the phase of being weakly efficient (Urquhart, 2016; Nadarajah & Chu, 2017) in the sense of the market efficiency hypothesis (Mantegna & Stanley, 2000). On the other hand, Bitcoin market exhibits speculative bubbles (Shiller, 2014; Cheah & Fry, 2015; Cheung et al., 2015). Its high volatility and substantial speculative component are related to the fact that Bitcoin is still more a trading asset than a currency (Dyhrberg, 2016; Katsiampa, 2017). The Bitcoin exchange rates have been already modeled and predicted by means of a noncausal autoregressive process (Hencic & Gourieroux, 2015). On the other hand, financial time ´ series, e.g. stocks, exchange rates, bonds or commodities, are characterised by the Hurst exponent (Liu et al., 1999; Carbone et al., 2004; Di Matteo et al., 2005; Sanchez Granero et al., ´ 2008; Zunino et al., 2017), and share the scaling properties with the fractional Brownian motion (fBm) (Lo, 1991; Cheridito, 2003; Bayraktar et al., 2004; Cont, 2005; Gu et al., 2012; Areerak, 2014; Vukovic, 2015). Therefore, in this work Bitcoin price forecasts are done via Monte Carlo simulations of a large sample of fBms. The most probable prices are then derived from the empirical cumulative distribution function (CDF). This paper is organised in the following manner. In Sect. 2 the data set and methods are described. Sect. 3 presents the results of the analysis, and in Sect. 4 discussion and concluding remarks are gathered.
2. Data and method
Data. The examined data set spans from 2011-12-18 to 2017- 07-06. The daily weighted prices were downloaded from BitcoinCharts.1 A three-day gap from 2015-01-06 to 2015-01-08 was linearly interpolated based on prices on 2015-01-05 and 2015-01-09. The total number of data points is 2028. The data is displayed in Fig. 1. Hurst exponent. To compute the Hurst exponent H, the wavelet approach using the Haar wavelet as a basis is employed (Tarnopolski, 2016); the error δH is the standard error of the slope of the fitted linear dependence of the logarithmic variance of the wavelet coefficients’ versus octave (see Tarnopolski (2016) and references therein for details). In order to make the size a power of 2, the examined data sets were padded left with zeros. For the whole dataset, H = 0.56 ± 0.07. Fig. 1 presents also the time evolution of H with a sliding window of size w = 512 and w = 1024, advancing by 1 data point. The w = 512 temporal evolution shows a roughly periodic pattern with decaying amplitude, with an outburst in the last ∼ 550 days above the plateau level. The picture drawn by the w = 1024 window is initially much more repetitive, with a prominent rise in the periods starting ∼ 1100 days ago. In both cases, the outbreak of H can be associated with the rally in the last weeks, during which the price increased by a factor of 2−3.
Methodology. The idea exploited herein is to generate a large number (105 ) of fractional Brownian motion (fBm) realisations with Hurst exponent H, and treat them as future price predictions, X(t). Two selection criteria are employed: (i) the price cannot drop below zero, and (ii) the value of H of the whole time series (i.e., the historical data and the generated fBm joined together) must lie within the interval H ± δH of the historical prices (i.e., an H constant in time is assumed). The fBms, B H t , are generated with drift µ and volatility σ the same as the historical prices yield (also assumed to be constant): X(t) = X0 1 + B H t (µ, σ) , where X0 is the initial price of the extensions, taken as the last price in the historical data set, and B H 0 = 0. From the selected realisations, an empirical cumulative distribution function (CDF), FX(x), is constructed and probabilities P of reaching prices X lower than x: P(X ≤ x) = FX(x), or higher than x: P(X > x) = 1 − FX(x), are computed. The approach is first tested on a set of prices til the end of 2016 (i.e., discarding the last 187 values and padding to the left with zeros), and then a prediction is made for the future 180 days. The computer algebra system Mathematica is applied throughout this paper; in particular, fBms are generated as FractionalBrownianMotionProcess[µ,σ,H].
3. Results
First, the set of prices til the end of 2016 is examined. It is characterised by H = 0.5±0.1 and consists of 1841 data points. Among the 105 realisations of fBm (of length 207 to eventually form a 211 = 2048 long time series), 72.2% met the selection criteria.2 Fig. 2 displays the results in its left column. The histogram is the empirical probability density function (PDF) of prices taken as the last value of the generated fBms (i.e., 2048 days after 2011-12-18, or on the 207th day of 2017), overlaid with a Gaussian fit which yielded a mean of 1677.95 USD. A variety of statistical tests (Kolmogorov-Smirnov, Cramer-von ´ Mises, Kuiper etc.) imply that a Gaussian distribution is a good description of the PDF. The probabilities of attaining a given price is calculated via the CDF; 955.73 USD is the price on 2016-12-31, and there was a 6.4% chance of reaching a price smaller than this. Exceeding the mean 1677.95 USD had a 53.6% probability, a threshold of 2000 USD or more yields a 24.6%, and rising above that last historical price of 2575.9 USD had a 0.6% chance of occurring. The middle column of Fig. 2 is the same as the left one, but the second selection criterion is such that H should fall within 0.56 ± 0.07, i.e. the value for the whole historical data set. This is performed as the value of H increased in the last months, as seen on Fig. 1, and hence the previous mean value of 1677.95 USD was underestimated. Employing the actual H yields a mean price of 1920.74 USD among the 76.5% of fBm realisations that met the selection criteria (the Gaussian distribution again passed all statistical tests). The probability of falling below 955.73 USD is now 0.7%, exceeding the mean is 50.2%, and achieving the actual price of 2575.9 USD has a 4.9% chance. Overall, the predicted price is 20% lower than the attained one, being a good agreement given the recent sudden rise in price, which lead to an increase of the instantaneous value of H. To make predictions about the Bitcoin price 180 days from 2017-06-07, again 105 fBm realisations were generated. The 2228 points long time series were padded left with zeros to obtain a 212 = 4096 long data sets. Among them, 62.5% met the selection criteria. The results are displayed in the right column of Fig. 2. The mean of the PDF was 4537.23 USD, i.e. this is the most probable price to be attained in the beginning of 2018. The probability of falling below the current price of 25575.9 USD is 6.3%, of exceeding the mean is 51.3%, and reaching a price higher than 8000 USD has a chance of 0.2%.
4. Concluding remarks
The Bitcoin price was modeled as an fBm, and price predictions were put forward through a Monte Carlo approach with 105 realisations. The predicted mid-2017 price, based on historical values til the end of 2016, was underestimated. However, using the most recent value of Hurst exponent, the prediction was lower than the value reached in relity by only 20%, which is considered as good agreement. Therefore, price predictions for the beginning of 2018 were made in the same way. It is found, via the CDF, that the most probable price (the mean of a normally distributed set of realisations) is 4537 USD, with a probability of exceeding this value equal to 51.3%. On the other hand, the chance of falling below the current price of 2575.9 USD is 6.3%.
It is well-known that large fluctuations in the market are more common than fBm predicts. In fact, extreme events—like the 1987’s Black Monday or the 2008-9 market crash—are virtually impossible under usual assumptions (Jackwerth & Rubinstein, 1996). This is true for the classical Black & Scholes (1973) model as well. A possible improvement might be to account for time variation of the volatility (Areerak, 2014) and the Hurst exponent. On the other hand, other models that incorporate bullish and bearish parameters to account for big upward and downward jumps, e.g. (Camara & Heston, 2008), are per- ˆ spective. Nevertheless, Bitcoin is still very young, giving a unique opportunity to observe an emerging cryptocurrency market, and potentially trigger new theories. Disclaimer. This manuscript is for information and illustrative purposes only. It is not, and should not be regarded as investment advice or as a recommendation regarding a course of action. The reader will make their own independent decision with respect to any course of action in connection herewith, as to whether such course of action is appropriate or proper based on their own judgment and their specific circumstances and objectives. The reader should seek a duly licensed professional for investment advice.
References
Ali, R., Barrdear, J., Clews, R., & Southgate, J. (2014). The economics of digital currencies. Bank of England Quarterly Bulletin, 54, 276–286. URL: http://EconPapers.repec.org/RePEc:boe:qbullt:0148. Areerak, T. (2014). Mathematical model of stock prices via a fractional brownian motion model with adaptive parameters. ISRN Applied Mathematics, 2014, 791418. URL: http://dx.doi.org/10.1155/2014/791418. doi:10.1155/2014/791418. Baek, C., & Elbeck, M. (2015). Bitcoins as an investment or speculative vehicle? a first look. Applied Economics Letters, 22, 30–34. URL: http://dx. doi.org/10.1080/13504851.2014.916379. doi:10.1080/13504851. 2014.916379. Bayraktar, E., Poor, H. V., & Sircar, K. R. (2004). Estimating the fractal dimension of the s&p index using wavelet analysis. International Journal of Theoretical and Applied Finance, 07, 615– 643. URL: http://www.worldscientific.com/doi/abs/10.1142/ S021902490400258X. doi:10.1142/S021902490400258X.
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–54. URL: http://www. jstor.org/stable/1831029. Bouri, E., Molnar, P., Azzi, G., Roubaud, D., & Hagfors, L. I. ´ (2017). On the hedge and safe haven properties of bitcoin: Is it really more than a diversifier? Finance Research Letters, 20, 192 – 198. URL: http://www.sciencedirect.com/science/ article/pii/S1544612316301817. doi:https://doi.org/10.1016/ j.frl.2016.09.025. Bradbury, D. (2013). The problem with bitcoin. Computer Fraud & Security, 2013, 5 – 8. URL: http://www.sciencedirect. com/science/article/pii/S1361372313701015. doi:http: //dx.doi.org/10.1016/S1361-3723(13)70101-5. Briere, M., Oosterlinck, K., & Szafarz, A. (2015). Virtual currency, tangible ` return: Portfolio diversification with bitcoin. Journal of Asset Management, 16, 365–373. URL: http://dx.doi.org/10.1057/jam.2015.5. doi:10.1057/jam.2015.5. Camara, A., & Heston, S. L. (2008). Closed-form option pricing formulas with ˆ extreme events. Journal of Futures Markets, 28, 213–230. URL: http: //dx.doi.org/10.1002/fut.20298. doi:10.1002/fut.20298. Carbone, A., Castelli, G., & Stanley, H. E. (2004). Time-dependent Hurst exponent in financial time series. Physica A Statistical Mechanics and its Applications, 344, 267–271. URL: http://www.sciencedirect.com/ science/article/pii/S0378437104009471. doi:10.1016/j.physa. 2004.06.130. Cheah, E.-T., & Fry, J. (2015). Speculative bubbles in bitcoin markets? an empirical investigation into the fundamental value of bitcoin. Economics Letters, 130, 32 – 36. URL: http://www. sciencedirect.com/science/article/pii/S0165176515000890. doi:http://dx.doi.org/10.1016/j.econlet.2015.02.029. Cheridito, P. (2003). Arbitrage in fractional brownian motion models. Finance and Stochastics, 7, 533–553. URL: http://dx.doi.org/10. 1007/s007800300101. doi:10.1007/s007800300101. Cheung, A. W.-K., Roca, E., & Su, J.-J. (2015). Crypto-currency bubbles: an application of the phillipsshiyu (2013) methodology on mt. gox bitcoin prices. Applied Economics, 47, 2348–2358. URL: http://dx.doi.org/10.1080/00036846.2015.1005827. doi:10.1080/00036846.2015.1005827. Cont, R. (2005). Long range dependence in financial markets. In J. Levy-V ´ ehel, & E. Lutton (Eds.), ´ Fractals in Engineering: New Trends in Theory and Applications (pp. 159–179). London: Springer London. URL: http://dx.doi.org/10.1007/1-84628-048-6_11. doi:10.1007/1-84628-048-6_11. Di Matteo, T., Aste, T., & Dacorogna, M. M. (2005). Long-term memories of developed and emerging markets: Using the scaling analysis to characterize their stage of development. Journal of Banking & Finance, 29, 827 – 851. URL: http://www. sciencedirect.com/science/article/pii/S0378426604001438. doi:http://dx.doi.org/10.1016/j.jbankfin.2004.08.004. Dwyer, G. P. (2015). The economics of bitcoin and similar private digital currencies. Journal of Financial Stability, 17, 81 – 91. URL: http://www.sciencedirect.com/science/article/ pii/S1572308914001259. doi:http://dx.doi.org/10.1016/j.jfs. 2014.11.006. Dyhrberg, A. H. (2016). Bitcoin, gold and the dollar – a garch volatility analysis. Finance Research Letters, 16, 85 – 92. URL: http://www. sciencedirect.com/science/article/pii/S1544612315001038. doi:http://dx.doi.org/10.1016/j.frl.2015.10.008. Gu, H., Liang, J.-R., & Zhang, Y.-X. (2012). Time-changed geometric fractional brownian motion and option pricing with transaction costs. Physica A: Statistical Mechanics and its Applications, 391, 3971 – 3977. URL: http://www.sciencedirect. com/science/article/pii/S0378437112002567. doi:http: //dx.doi.org/10.1016/j.physa.2012.03.020. Hencic, A., & Gourieroux, C. (2015). Noncausal autoregressive model in ap- ´ plication to bitcoin/usd exchange rates. In V.-N. Huynh, V. Kreinovich, S. Sriboonchitta, & K. Suriya (Eds.), Econometrics of Risk (pp. 17–40). Springer International Publishing. URL: http://dx.doi.org/10.1007/ 978-3-319-13449-9_2. doi:10.1007/978-3-319-13449-9_2. Jackwerth, J., & Rubinstein, M. (1996). Recovering probability distributions from option prices. Journal of Finance, 51, 1611–32. URL: http://www.jstor.org/stable/2329531. doi:10.2307/2329531. Katsiampa, P. (2017). Volatility estimation for bitcoin: A comparison of garch models. Economics Letters, 158, 3 – 6. URL: http://www. sciencedirect.com/science/article/pii/S0165176517302501. doi:https://doi.org/10.1016/j.econlet.2017.06.023. Liu, Y., Gopikrishnan, P., Cizeau, P., Meyer, M., Peng, C.-K., & Stanley, H. E. (1999). Statistical properties of the volatility of price fluctuations. Phys. Rev. E, 60, 1390–1400. URL: https://link.aps.org/doi/10.1103/ PhysRevE.60.1390. doi:10.1103/PhysRevE.60.1390. Lo, A. W. (1991). Long-term memory in stock market prices. Econometrica, 59, 1279–1313. URL: http://www.jstor.org/stable/2938368. doi:10.2307/2938368. Mantegna, R. N., & Stanley, H. E. (2000). An Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge University Press, New York. Nadarajah, S., & Chu, J. (2017). On the inefficiency of bitcoin. Economics Letters, 150, 6 – 9. URL: http://www.sciencedirect.com/science/ article/pii/S0165176516304426. doi:https://doi.org/10.1016/ j.econlet.2016.10.033. Nakamoto, S. (2009). Bitcoin: A peer-to-peer electronic cash system, . URL: https://bitcoin.org/bitcoin.pdf/. Sanchez Granero, M. A., Trinidad Segovia, J. E., & Garc ´ ´ıa Perez, J. ´ (2008). Some comments on hurst exponent and the long memory processes on capital markets. Physica A: Statistical Mechanics and its Applications, 387, 5543 – 5551. URL: http://www. sciencedirect.com/science/article/pii/S0378437108004895. doi:http://dx.doi.org/10.1016/j.physa.2008.05.053. Shiller, R. J. (2014). Speculative asset prices. American Economic Review, 104, 1486–1517. URL: http://www.aeaweb.org/articles?id=10.1257/ aer.104.6.1486. doi:10.1257/aer.104.6.1486. Tarnopolski, M. (2016). On the relationship between the hurst exponent, the ratio of the mean square successive difference to the variance, and the number of turning points. Physica A: Statistical Mechanics and its Applications, 461, 662 – 673. URL: http://www. sciencedirect.com/science/article/pii/S0378437116302722. doi:http://dx.doi.org/10.1016/j.physa.2016.06.004. Urquhart, A. (2016). The inefficiency of bitcoin. Economics Letters, 148, 80 – 82. URL: http://www.sciencedirect.com/science/article/pii/ S0165176516303640. doi:http://dx.doi.org/10.1016/j.econlet. 2016.09.019. Vranken, H. (2017). Sustainability of bitcoin and blockchains. Current Opinion in Environmental Sustainability, 28, 1 – 9. URL: http://www. sciencedirect.com/science/article/pii/S1877343517300015. doi:https://doi.org/10.1016/j.cosust.2017.04.011. Vukovic, O. (2015). Analysing the chinese stock market using the hurst exponent, fractional brownian motion and variants of a stochastic logistic differential equation. International Journal of Design & Nature and Ecodynamics, 10, 300–309. URL: https://www.witpress.com/elibrary/ dne-volumes/10/4/1010. doi:10.2495/DNE-V10-N4-300-309. Zunino, L., Olivares, F., Bariviera, A. F., & Rosso, O. A. (2017). A simple and fast representation space for classifying complex time series. Physics Letters A, 381, 1021 – 1028. URL: http://www.sciencedirect.com/ science/article/pii/S0375960116316681. doi:https://doi.org/ 10.1016/j.physleta.2017.01.047.
https://arxiv.org/pdf/1707.03746.pdf