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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Complex Dynamics and Nonlinear Interactions in Bitcoin Price Modeling

Discontinuity, Nonlinearity, and Complexity 15(4) (2026) 463--490 | DOI:10.5890/DNC.2026.12.001

Jules Clement Mba, Gomolemo Goodwill Motloba, Abdulrazak Abdulrahman Abubakar

School of Economics, University of Johannesburg, P. O. Box 524 Auckland, 2006 Johannesburg, South Africa

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Abstract

This paper develops a mathematical framework for modeling Bitcoin price dynamics through a system of coupled stochastic differential equations (SDEs). We capture the complex nonlinear interactions between Bitcoin price and five key factors: investor sentiment, trading volume, mining hashrate, transaction fees, and transaction counts. The model incorporates jump processes to account for sudden price movements and regime-switching to capture state-dependent dynamics. We derive the resulting partial differential equations for derivative pricing and analyze the system's behavior through simulation. Our empirical findings suggest significant feedback mechanisms between network metrics and price dynamics, with hashrate exhibiting the strongest correlation with price movements. The framework provides a foundation for understanding the complex, non-linear, and fractal-like behavior observed in cryptocurrency markets while enabling the pricing of derivatives in this emerging asset class.

References

  1. [1]  Nakamoto, S. (2008), Bitcoin: A peer-to-peer electronic cash system, Available at: https://bitcoin.org/bitcoin.pdf.
  2. [2]  Liu, Y. and Tsyvinski, A. (2021), Risks and returns of cryptocurrency, The Review of Financial Studies, 34(6), 2689-2727.
  3. [3]  Pagnotta, E. and Buraschi, A. (2018), An equilibrium valuation of Bitcoin and decentralized network assets, Journal of Finance, 1-40.
  4. [4]  Baur, D.G., Hong, K., and Lee, A.D. (2018), Bitcoin: Medium of exchange or speculative assets?, Journal of International Financial Markets, Institutions and Money, 54, 177-189.
  5. [5]  Dyhrberg, A.H. (2016), Bitcoin, gold and the dollar--A GARCH volatility analysis, Finance Research Letters, 16, 85-92.
  6. [6]  Chan, S., Chu, J., Nadarajah, S., and Osterrieder, J. (2017), A statistical analysis of cryptocurrencies, Journal of Risk and Financial Management, 10(2), 12.
  7. [7]  Dyhrberg, A.H. (2016), Hedging capabilities of bitcoin: Is it the virtual gold?, Finance Research Letters, 16, 139-144.
  8. [8]  Abubakar, A.A., Mba, J.C., and Ohonba, A. (2024), Analysis of tail dependence structure and risk spillover between cryptocurrencies, Investment Management & Financial Innovations, 21(4), 140-155.
  9. [9]  Katsiampa, P. (2017), Volatility estimation for Bitcoin: A comparison of GARCH models, Economics Letters, 158, 3-6.
  10. [10]  Kristoufek, L. (2015), What are the main drivers of the Bitcoin price? Evidence from wavelet coherence analysis, PLoS One, 10(4), e0123923.
  11. [11]  Cong, L.W., Li, Y., and Wang, N. (2021), Tokenomics: Dynamic adoption and valuation, Review of Financial Studies, 34(3), 1105-1155.
  12. [12]  Chen, C.Y.H., Härdle, W.K., Hou, A.J., and Wang, W. (2020), Pricing cryptocurrency options: The case of CRIX and Bitcoin, Journal of Financial Econometrics, 18(2), 250-279.
  13. [13]  Gronwald, M. (2019), Is Bitcoin a commodity? On price jumps, demand shocks, and certainty of supply, Journal of International Money and Finance, 97, 86-92.
  14. [14]  Scaillet, O., Treccani, A., and Trevisan, C. (2020), High-frequency jump analysis of the bitcoin market, Journal of Financial Econometrics, 18(2), 209-232.
  15. [15]  Hou, A.J., Wang, W., Chen, C.Y.H., and Härdle, W.K. (2021), Pricing cryptocurrency options, Journal of Financial Econometrics, 19(1), 34-67.
  16. [16]  Cocco, L., Concas, G., and Marchesi, M. (2017), Using an artificial financial market for studying a cryptocurrency market, Journal of Economic Interaction and Coordination, 12(2), 345-365.
  17. [17]  Jang, H. and Lee, J. (2018), An empirical study on modeling and prediction of bitcoin prices with bayesian neural networks based on blockchain information, IEEE Access, 6, 5427-5437.
  18. [18]  McNally, S., Roche, J., and Caton, S. (2018), Predicting the price of Bitcoin using machine learning, In 26th Euromicro International Conference on Parallel, Distributed and Network-based Processing (PDP), 339-343, IEEE.
  19. [19]  Takaishi, T. (2018), Statistical properties and multifractality of Bitcoin, Physica A: Statistical Mechanics and its Applications, 506, 507-519.
  20. [20]  Heston, S.L. (1993), A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6(2), 327-343.
  21. [21]  Merton, R.C. (1976), Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3(1-2), 125-144.
  22. [22]  Filimonov, V. and Sornette, D. (2015), Apparent criticality and calibration issues in the Hawkes self-excited point process model: Application to high-frequency financial data, Quantitative Finance, 15(8), 1293-1314.
  23. [23]  Mensi, W., Al-Yahyaee, K.H., and Kang, S.H. (2019), Structural breaks and double long memory of cryptocurrency prices: A comparative analysis from Bitcoin and Ethereum, Finance Research Letters, 29, 222-230.
  24. [24]  Gardiner, C.W. (1985), Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer-Verlag: Berlin.
  25. [25]  Khasminskii, R. (2011), Stochastic Stability of Differential Equations, Vol. 66, Springer Science & Business Media.
  26. [26]  Aït-Sahalia, Y. and Jacod, J. (2007), Volatility estimators for discretely sampled Lévy processes, The Annals of Statistics, 35(1), 355-392.
  27. [27]  Corbet, S., Lucey, B., Urquhart, A., and Yarovaya, L. (2019), Cryptocurrencies as a financial asset: A systematic analysis, International Review of Financial Analysis, 62, 182-199.
  28. [28]  Lischke, M. and Fabian, B. (2016), Analyzing the Bitcoin network: The first four years, Future Internet, 8(1), 7.
  29. [29]  Liu, Y. and Tsyvinski, A. (2021), Risks and returns of cryptocurrency, The Review of Financial Studies, 34(6), 2689-2727.
  30. [30]  Scaillet, O., Treccani, A., and Trevisan, C. (2020), High-frequency jump analysis of the bitcoin market, Journal of Financial Econometrics, 18(2), 209-232.
  31. [31]  Cong, L.W., Li, Y., and Wang, N. (2021), Tokenomics: Dynamic adoption and valuation, Review of Financial Studies, 34(3), 1105-1155.

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