Bitcoin volatility forecasting: An artificial differential equation neural network

S. Pourmohammad Azizi; Chien Yi Huang; Ti An Chen; Shu Chuan Chen; Amirhossein Nafei; S. Pourmohammad Azizi; Chien Yi Huang; Ti An Chen; Shu Chuan Chen; Amirhossein Nafei

doi:10.3934/math.2023712

AIMS Mathematics

2023, Volume 8, Issue 6: 13907-13922. doi: 10.3934/math.2023712

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Research article Special Issues

Bitcoin volatility forecasting: An artificial differential equation neural network

1.
Department of Electrical Engineering, National Taiwan Ocean University, Keelung, Taiwan
2.
Department of Industrial Engineering and Management, National Taipei University of Technology, Taipei, Taiwan
3.
Department of Business Administration, JinWen University of Science Technology, Taipei, Taiwan
4.
Department of Mathematics, Semnan University, Semnan, Iran

Received: 07 January 2023 Revised: 16 March 2023 Accepted: 26 March 2023 Published: 13 April 2023
MSC : 91-08, 68T01

In this article, an alternate method for estimating the volatility parameter of Bitcoin is provided. Specifically, the procedure takes into account historical data. This quality is one of the most critical factors determining the Bitcoin price. The reader will notice an emphasis on historical knowledge throughout the text, with particular attention paid to detail. Following the production of a historical data set for volatility utilizing market data, we will analyze the fundamental and computed values of Bitcoin derivatives (futures), followed by implementing an inverse problem modeling method to obtain a second-order differential equation model for volatility. Because of this, we can accomplish what we set out to do. As a direct result, we will be able to achieve our objective. Following this, the differential equation of the second order will be solved by an artificial neural network that considers the dataset. In conclusion, the results achieved through the utilization of the Python software are given and contrasted with a variety of other research approaches. In addition, this method is determined with alternative ways, and the outcomes of those comparisons are shown.
- Bitcoin,
- volatility,
- differential equation,
- artificial neural network,
- forecasting,
- inverse problem
Citation: S. Pourmohammad Azizi, Chien Yi Huang, Ti An Chen, Shu Chuan Chen, Amirhossein Nafei. Bitcoin volatility forecasting: An artificial differential equation neural network[J]. AIMS Mathematics, 2023, 8(6): 13907-13922. doi: 10.3934/math.2023712

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Abstract

In this article, an alternate method for estimating the volatility parameter of Bitcoin is provided. Specifically, the procedure takes into account historical data. This quality is one of the most critical factors determining the Bitcoin price. The reader will notice an emphasis on historical knowledge throughout the text, with particular attention paid to detail. Following the production of a historical data set for volatility utilizing market data, we will analyze the fundamental and computed values of Bitcoin derivatives (futures), followed by implementing an inverse problem modeling method to obtain a second-order differential equation model for volatility. Because of this, we can accomplish what we set out to do. As a direct result, we will be able to achieve our objective. Following this, the differential equation of the second order will be solved by an artificial neural network that considers the dataset. In conclusion, the results achieved through the utilization of the Python software are given and contrasted with a variety of other research approaches. In addition, this method is determined with alternative ways, and the outcomes of those comparisons are shown.

References

[1]	A. H. Dyhrberg, Hedging capabilities of bitcoin, Is it the virtual gold? Financ. Res. Lett., 16 (2016), 139–144. https://doi.org/10.1016/j.frl.2015.10.025
[2]	T. G. Andersen, T. G. Bollerslev, F. X. Diebold, P. Labys, Modeling and forecasting realized volatility, Econometrica, 71 (2003), 579–625. https://doi.org/10.1111/1468-0262.00418 doi: 10.1111/1468-0262.00418
[3]	T. G. Andersen, T. Bollerslev, N. Meddahi, Correcting the errors: Volatility forecast evaluation using high-frequency data and realized volatilities, Econometrica, 73 (2005), 279–296. https://doi.org/10.1111/j.1468-0262.2005.00572.x doi: 10.1111/j.1468-0262.2005.00572.x
[4]	S. P. Azizi, A. Neisy, A new approach in geometric brownian motion model, In: Cao, BY. Eds, Fuzzy Information and Engineering and Decision, IWDS 2016, Adv. Intell. Syst. Comput., 646 (2018), 336–342. https://doi.org/10.1007/978-3-319-66514-6_34 doi: 10.1007/978-3-319-66514-6_34
[5]	W. Bao, J. Yue, Y. Rao, A deep learning framework for financial time series using stacked autoencoders and long-short term memory, PloS One, 12 (2017), e0180944. https://doi.org/10.1371/journal.pone.0180944 doi: 10.1371/journal.pone.0180944
[6]	R. Becker, A. E. Clements, M. B. Doolan, A. S. Hurn, Selecting volatility forecasting models for portfolio allocation purposes, Int. J. Forecasting, 31 (2015), 849–861. https://doi.org/10.1016/j.ijforecast.2013.11.007 doi: 10.1016/j.ijforecast.2013.11.007
[7]	T. Björk, Arbitrage theory in continuous time, Oxford university press, 2009. https://doi.org/10.1093/0199271267.001.0001
[8]	F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637–654. https://www.jstor.org/stable/1831029
[9]	T. Bollerslev, B. Hood, J. Huss, L. H. Pedersen, Risk everywhere: Modeling and managing volatility, Rev. Financ. Stud., 31 (2018), 2729–2773. http://dx.doi.org/10.2139/ssrn.2722591 doi: 10.2139/ssrn.2722591
[10]	T. Bollerslev, A. J. Patton, R. Quaedvlieg, Exploiting the errors: A simple approach for improved volatility forecasting, J. Econ., 192 (2016), 1–18. https://doi.org/10.1016/j.jeconom.2015.10.007 doi: 10.1016/j.jeconom.2015.10.007
[11]	J. Bouoiyour, R. Selmi, Bitcoin price: Is it really that new round of volatility can be on way?, University Library of Munich, Germany, (2015). Available from: https://mpra.ub.uni-muenchen.de/65580/
[12]	J. Bouoiyour, R. Selmi, Bitcoin: A beginning of a new phase, Econ. B., 36 (2016), 1430–1440. Available from: https://econpapers.repec.org/article/eblecbull/eb-16-00372.htm
[13]	E. Bouri, G. Azzi, A. H. Dyhrberg, On the return-volatility relationship in the Bitcoin market around the price crash of 2013, Economics, 11 (2017). https://doi.org/10.5018/economics-ejournal.ja.2017-2
[14]	Á. Cebrián-Hernández, E. Jiménez-Rodríguez, Modeling of the bitcoin volatility through key financial environment variables: An application of conditional correlation MGARCH models, Mathematics, 9 (2021), 267. https://doi.org/10.3390/math9030267 doi: 10.3390/math9030267
[15]	A. Charles, O. Darné, Volatility estimation for Bitcoin: Replication and robustness, Int. Econ., 157 (2019), 23–32. https://doi.org/10.1016/j.inteco.2018.06.004 doi: 10.1016/j.inteco.2018.06.004
[16]	C. Chiarella, M. Craddock, N. El-Hassan, The calibration of stock option pricing models using inverse problem methodology, QFRQ Research Papers, UTS Sydney, 39 (2000). Available from: https://ideas.repec.org/p/uts/rpaper/39.html
[17]	F. Corsi, A simple approximate long-memory model of realized volatility, J. Financ. Econ., 7 (2009), 174–196. https://doi.org/10.1093/jjfinec/nbp001 doi: 10.1093/jjfinec/nbp001
[18]	F. Corsi, D. Pirino, R. Reno, Threshold bipower variation and the impact of jumps on volatility forecasting, J. Econ., 159 (2010), 276–288. https://doi.org/10.1016/j.jeconom.2010.07.008 doi: 10.1016/j.jeconom.2010.07.008
[19]	G. Cortazar, E. S. Schwartz, Implementing a stochastic model for oil futures prices, Energy Econ., 25 (2003), 215–238. https://doi.org/10.1016/S0140-9883(02)00096-8 doi: 10.1016/S0140-9883(02)00096-8
[20]	P. Della Corte, T. Ramadorai, L. Sarno, Volatility risk premia and exchange rate predictability, J. Financ. Econ., 120 (2016), 21–40. https://doi.org/10.1016/j.jfineco.2016.02.015 doi: 10.1016/j.jfineco.2016.02.015
[21]	C. Dritsaki, An empirical evaluation in GARCH volatility modeling: Evidence from the Stockholm stock exchange, J. Math. Financ., 7 (2017), 366–390. https://doi.org/10.4236/jmf.2017.72020 doi: 10.4236/jmf.2017.72020
[22]	D. J. Duffy, Finite Difference methods in financial engineering: A Partial Differential Equation approach, John Wiley & Sons, 2013. https://doi.org/10.1002/9781118673447
[23]	A. H. Dyhrberg, Bitcoin, gold and the dollar–A GARCH volatility analysis, Financ. Res. Lett., 16 (2016), 85–92. https://doi.org/10.1016/j.frl.2015.10.008 doi: 10.1016/j.frl.2015.10.008
[24]	R. F. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica: J. Econ. Soc., 50 (1982), 987–1007. https://doi.org/10.2307/1912773 doi: 10.2307/1912773
[25]	C. Francq, J. M. Zakoian, GARCH models: Structure, statistical inference and financial applications, John Wiley & Sons, 2019. ISBN: 978-1-119-31348-9.
[26]	G. M. Gallo, E. Otranto, Forecasting realized volatility with changing average levels, Int. J. Forecasting, 31 (2015), 620–634. https://doi.org/10.1016/j.ijforecast.2014.09.005 doi: 10.1016/j.ijforecast.2014.09.005
[27]	F. Glaser, K. Zimmermann, M. Haferkorn, M. C. Weber, M. Siering, Bitcoin-asset or currency? revealing users' hidden intentions, Revealing Users' Hidden Intentions, (2014) ECIS. Available from: https://ssrn.com/abstract = 2425247
[28]	X. Gong, B. Lin, Structural breaks and volatility forecasting in the copper futures market, J. Futures Markets, 38 (2018), 290–339. https://doi.org/10.1002/fut.21867 doi: 10.1002/fut.21867
[29]	C. Gouriéroux, ARCH models and financial applications, Springer Science & Business Media (2012). https://doi.org/10.1007/978-1-4612-1860-9
[30]	A. Graves, Supervised Sequence Labelling with Recurrent Neural Networks, Springer Berlin, Heidelberg (2012). https://doi.org/10.1007/978-3-642-24797-2
[31]	M. Grigoriu, Stochastic calculus: applications in science and engineering, Birkhäuser Boston, MA, Springer Science & Business Media (2013). https://doi.org/10.1007/978-0-8176-8228-6
[32]	M. Gronwald, The Economics of Bitcoins – Market Characteristics and Price Jumps, CESifo Working Paper Series, 5121 (2014). http://dx.doi.org/10.2139/ssrn.2548999
[33]	T. Guo, A. Bifet, N. Antulov-Fantulin, Bitcoin Volatility Forecasting with a Glimpse into Buy and Sell Orders, 2018 IEEE International Conference on Data Mining (ICDM), Singapore, (2018), 989–994, http://dx.doi:10.1109/ICDM.2018.00123
[34]	L. T. Hoang, D. G. Baur, Forecasting bitcoin volatility: Evidence from the options market, J. Futures Markets, 40 (2020), 1584–1602. https://doi.org/10.1002/fut.22144 doi: 10.1002/fut.22144
[35]	J. N. Hugonnier, The Feynman–Kac formula and pricing occupation time derivatives, Int. J. Theor. Appl. Fin., 2 (1999), 153–178. https://doi.org/10.1142/S021902499900011X doi: 10.1142/S021902499900011X
[36]	S. Karaoglu, U. Arpaci, S. Ayvaz, A deep learning approach for optimization of systematic signal detection in financial trading systems with big data, Int. J. Intell. Syst. Appl. Eng., 7 (20119), 31–36. https://doi.org/10.18201/ijisae.2017SpecialIssue31421
[37]	P. Katsiampa, Volatility estimation for Bitcoin: A comparison of GARCH models, Econ. Lett., 158 (2017), 3–6. https://doi.org/10.1016/j.econlet.2017.06.023 doi: 10.1016/j.econlet.2017.06.023
[38]	A. Kirsch, An introduction to the mathematical theory of inverse problems, Springer New York, NY (2011). https://doi.org/10.1007/978-1-4419-8474-6
[39]	F. C. Klebaner, Introduction to stochastic calculus with applications, World Scientific Publishing Company, (2012). https://doi.org/10.1142/p821
[40]	R. Lagnado, S. Osher, A technique for calibrating derivative security pricing models: numerical solution of an inverse problem, J. Comput. Finance, 1 (1997), 13–26. https://doi.org/10.1088/0266-5611/15/3/201 doi: 10.1088/0266-5611/15/3/201
[41]	B. Lei, Z. Liu, Y. Song, On Stock Volatility Forecasting Based on Text Mining and Deep Learning Under High Frequency Data, J. Forecasting, 40 (2021), 1596–1610. https://doi.org/10.1002/for.2794 doi: 10.1002/for.2794
[42]	G. Levy, Computational finance: numerical methods for pricing financial instruments, Butterworth-Heinemann, (2004). ISBN: 9780080472270.
[43]	F. Ma, Y. Wei, L. Liu, D. Huang, Forecasting realized volatility of oil futures market: A new insight, J. Forecasting, 37 (2018), 419–436. https://doi.org/10.1002/for.2511 doi: 10.1002/for.2511
[44]	R. C. Merton, An intertemporal capital asset pricing model, Econometrica: J. Econ. Soc., 41 (1973), 867–887. https://doi.org/10.2307/1913811 doi: 10.2307/1913811
[45]	A. Neisy, K. Salmani, An inverse finance problem for estimation of the volatility, Comp. Math. Math. Phys., 53 (2013), 63–77. https://doi.org/10.1134/S0965542513010090 doi: 10.1134/S0965542513010090
[46]	R. Paschke, M. Prokopczuk Commodity derivatives valuation with autoregressive and moving average components in the price dynamics, J. Banking Financ., 34 (2010), 2742–2752. https://doi.org/10.1016/j.jbankfin.2010.05.010 doi: 10.1016/j.jbankfin.2010.05.010
[47]	A. J. Patton, Volatility forecast comparison using imperfect volatility proxies, J. Econometrics, 160 (2011), 246–256. https://doi.org/10.1016/j.jeconom.2010.03.034 doi: 10.1016/j.jeconom.2010.03.034
[48]	S. H. Poon, C. W. Granger, Forecasting volatility in financial markets: A review, J. Econ. Lit., 41 (2003), 478–539. https://doi.org/10.1257/002205103765762743 doi: 10.1257/002205103765762743
[49]	S. Shreve, Stochastic calculus for finance I: the binomial asset pricing model, Springer Science Business Media, (2005). https://doi.org/10.1007/978-0-387-22527-2
[50]	S. E. Shreve, Stochastic calculus for finance II: Continuous-time models, Springer Science & Business Media, (2004). Available from: https://link.springer.com/book/9780387401010
[51]	The Euler-Lagrange Equation, https://www.ucl.ac.uk.
[52]	Y. Wang, H. Wang, S. Zhang, Prediction of daily PM2. 5 concentration in China using data-driven ordinary differential equations, Appl. Math. Comput., 375 (2020), 125088. https://doi.org/10.1016/j.amc.2020.125088 doi: 10.1016/j.amc.2020.125088
[53]	R. Weinstock, Calculus of variations: with applications to physics and engineering. Courier Corporation, McGraw-Hill, (1974). ISBN: 978-0486630694.
[54]	J. Wloka, Partial differential equations, Cambridge University Press, Cambridge, UK (1987). https://doi.org/10.1017/CBO9781139171755
[55]	Z. Xu, X. Jia, The calibration of volatility for option pricing models with jump diffusion processes, Appl. Anal., 98 (2017), 810–827. https://doi.org/10.1080/00036811.2017.1403588 doi: 10.1080/00036811.2017.1403588
[56]	Y. Yu, W. Duan, Q. Cao, The impact of social and conventional media on firm equity value: A sentiment analysis approach, Decis. Support Syst., 55 (2013), 919–926. https://doi.org/10.1016/j.dss.2012.12.028 doi: 10.1016/j.dss.2012.12.028

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