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A Fractional Calculus Approach to Modeling the Rehabilitation Dynamics of Stock Markets

Silas Abahia Ihedioha, Chinedu Bright Okechukwu, Onyekachi P. Ogwo, Chigozie Chibuisi, Bright Okore Osu

Abstract

The dynamics of stock markets often exhibit complex behaviors that traditional models struggle to capture, particularly in recovery phases following periods of volatility. This paper introduces a novel mathematical model based on fractional calculus to describe and predict the rehabilitation dynamics of stock markets. We formulate a fractional differential equation (FDE) model and validate its effectiveness using historical market data. The model's ability to account for long-term memory effects and non-local interactions offers significant advantages over classical approaches.

Keywords

Fractional calculusfractional differential equationsstock market dynamicshealing

References

[1] Barberis, N., & Thaler, R. (2003). A survey of behavioral finance. Handbook of the Economics of Finance, 1, 1053-1128. [2] Clarke, J., Cowan, A., & Naughton, T. (2018). Circuit breakers and the effectiveness of market regulations: Evidence from the 2015 Chinese stock market crash. Journal of Financial Markets, 39, 120-144. [3] Farhi, E., &Tirole, J. (2012). Collective moral hazard, maturity mismatch, and systemic bailouts. American Economic Review, 102(1), 60-93. [4] Fawley, B. W., & Neely, C. J. (2013). Four stories of quantitative easing. Federal Reserve Bank of St. Louis Review, 95(1), 51-88. [5] Menkveld, A. J. (2013). High-frequency trading and the new market makers. Journal of Financial Markets, 16(4), 712-740. [6] Barberis, N., & Thaler, R. (2003). A survey of behavioral finance. Handbook of the Economics of Finance, 1, 1053-1128. [7] Clarke, J., Cowan, A., & Naughton, T. (2018). Circuit breakers and the effectiveness of market regulations: Evidence from the 2015 Chinese stock market crash. Journal of Financial Markets, 39, 120-144. [8] Asness, C. S., Moskowitz, T. J., & Pedersen, L. H. (2013). Value and momentum everywhere. The Journal of Finance, 68(3), 929-985. [9] Barberis, N., Shleifer, A., & Vishny, R. (1998). A model of investor sentiment. Journal of Financial Economics, 49(3), 307-343. [10] Daniel, K., & Moskowitz, T. J. (2016). Momentum crashes. Journal of Financial Economics, 122(2), 221-247. [10] Jegadeesh, N., & Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock market efficiency. The Journal of Finance, 48(1), 65-91. [11] Moskowitz, T. J., Ooi, Y. H., & Pedersen, L. H. (2012). Time series momentum. Journal of Financial Economics, 104(2), 228-250. [12] Baker, M., & Wurgler, J. (2006). Investor sentiment and the cross-section of stock returns. The Journal of Finance, 61(4), 1645-1680. [13] Baker, M., & Wurgler, J. (2007). Investor sentiment in the stock market. Journal of Economic Perspectives, 21(2), 129-152. [14] Romer, C. D., & Romer, D. H. (2000). Federal Reserve information and the behavior of interest rates. American Economic Review, 90(3), 429-457. [15] Shiller, R. J. (2017). Narrative economics. American Economic Review, 107(4), 967-1004. [16] Tetlock, P. C. (2007). Giving content to investor sentiment: The role of media in the stock market. The Journal of Finance, 62(3), 1139-1168. [17] Whaley, R. E. (2000). The investor fear gauge. The Journal of Portfolio Management, 26(3), 12-17. [18] Bekaert, G., Hoerova, M., & Duca, M. L. (2013). Risk, uncertainty and monetary policy. Journal of Monetary Economics, 60(7), 771-788. [19] Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3(1- 2), 167-179. [20] Danielsson, J., Shin, H. S., & Zigrand, J. P. (2012). Endogenous risk and risk management. In Handbook of Systemic Risk (pp. 253-282). Cambridge University Press. [21] Harris, L. (1998). Circuit breakers and program trading limits: What have we learned? Brookings-Wharton Papers on Financial Services, 1998(1), 17-63. [22] Menkveld, A. J. (2013). High-frequency trading and the new market makers. Journal of Financial Markets, 16(4), 712-740. [23] Shiller, R. J. (1981). Do stock prices move too much to be justified by subsequent changes in dividends? American Economic Review, 71(3), 421-436. [24] Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1), 3-56. [25] Gopinath, G. (2020). The great lockdown: Dissecting the economic effects. IMF Economic Review, 68(2), 91-125. [26] Kindleberger, C. P., &Aliber, R. Z. (2011). Manias, Panics, and Crashes: A History of Financial Crises. Palgrave Macmillan. [27] Lo, A. W., &MacKinlay, A. C. (1990). When are contrarian profits due to stock market overreaction? The Review of Financial Studies, 3(2), 175-205. [28] Reinhart, C. M., & Rogoff, K. S. (2009). This Time Is Different: Eight Centuries of Financial Folly. Princeton University Press. [29] Shiller, R. J. (2003). From efficient markets theory to behavioral finance. Journal of Economic Perspectives, 17(1), 83-104. [30] Bernanke, B. S., & Gertler, M. (2001). Should central banks respond to movements in asset prices? American Economic Review, 91(2), 253-257. [31] Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-654. [32]Blinder, A. S. (2010). How central should the central bank be? Journal of Economic Literature, 48(1), 123-133. [33] Harris, L. (1998). Circuit breakers and program trading limits: What have we learned? Brookings-Wharton Papers on Financial Services, 1998(1), 17-63. [34] Madhavan, A., & Smidt, S. (1993). An analysis of changes in specialist inventories and quotations. Journal of Finance, 48(5), 1595-1628. [35] Shiller, R. J. (1981). Do stock prices move too much to be justified by subsequent changes in dividends? American Economic Review, 71(3), 421-436. [36] Stiglitz, J. E. (2010). Risk and global economic architecture: Why full financial integration may be undesirable. American Economic Review, 100(2), 388-392 [37] Adomian, G. (1994). Solving Frontier Problems of Physics: The Decomposition Method. Springer. [38] Wazwaz, A. M. (2009). Partial Differential Equations and Solitary Waves Theory. Springer.