Modelling the Fractional Diffusion Equation for a Truncated Levy Process with Financial Applications
DOI:
https://doi.org/10.62933/jypqf538Abstract
The truncated Levy process (TLP) modifies the heavy-tailed Levy distribution by transitioning to a fast-decaying probability distribution, solving the second-moment divergence problem. We present an extension of the fractional diffusion equation that simulates a process with a truncated Levy power-law behavior with an exponent of 5-α. This results in a closed-form discriminant function, where the displacement probability density function transitions to a Gaussian in essence while preserving the power-law tail. Truncated Levy processes are promising for financial modelling, as they provide finite moments and capture short-term divergence and long-term Gaussian convergence. We validate the truncated Levy process model using simulated data and then using Iraqi stock market data, exploring option hedging in a Levy-dominated context, comparing optimal strategies with delta hedging, and revealing key differences. In addition, we derive a generalized option pricing formula for assets under the truncated Levy process model
References
[1] Barkai, E., Metzler, R., & Osorio, M. (2012). From continuous time random walks to the fractional diffusion equation. Physical Review X, 2(4), 041010.
[2] Ben-Avraham, D., & Havlin, S. (2000). Diffusion and Reactions in Fractals and Disordered Systems. Springer.
[3] Bouchaud, J.-P., Gefen, Y., & Parisi, G. (2000). Universal features of Lévy walks and their applications to finance. Physical Review Letters, 87(22), 228701.
[4] Deng, X., et al. (2018). Anomalous diffusion in solid-state electrolytes: A review. Journal of Materials Chemistry A, 6(27), 12993-13008.
[5] Einstein, A. (1905). On the movement of small particles suspended in a stationary liquid. Annalen der Physik, 322(8), 549-560.
[6] Gao, Y., et al. (2020). Machine learning for anomalous diffusion. Nature Reviews Physics, 2(9), 466-479.
[7] Mandelbrot, B. B., & Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM review, 10(4), 422-437.
[8] Ghosh, A., et al. (2020). Anomalous diffusion in biological systems: A review. Journal of Physics: Condensed Matter, 32(12), 123001.
[9] Havlin, S., & Ben-Avraham, D. (2002). Diffusion in disordered media. Advances in Physics, 51(3), 187-292.
[10] Metzler, R., & Klafter, J. (2000). The random walk's guide to anomalous diffusion: A fractional dynamics approach. Physics Reports, 339(1), 1-77.
[11] Montroll, E. W., & Weiss, G. H. (1965). Random walks on lattices II. B. Continuous time random walks and optimal search processes. Journal of Mathematical Physics, 6(2), 167-181.
[12] Podlubny, I. (1999). Fractional Differential Equations. Academic Press.
[13] Scher, H., & Lax, M. (1973). Stochastic transport in a disordered medium. Physical Review B, 7(10), 4502-4510.
[14] Scher, H., et al. (2018). Anomalous diffusion in disordered media: A review of recent developments. Journal of Statistical Physics, 173(5), 1347-1376.
[15] Viswanathan, G. M., et al. (1999). Optimizing the encounter rate in biological interactions: The Lévy walk. Physical Review E, 61(6), 5606-5612.
[16] Weiss, M., et al. (2004). Anomalous subdiffusion can be explained by a combination of hindrance and trapping. Physical Review Letters, 93(2), 020601.
[17] Yuan, H., et al. (2016). Anomalous diffusion in heterogeneous media: A review. Journal of Physics: Condensed Matter, 28(20), 203001.
[18] Zhang, H., et al. (2017). Tailoring anomalous diffusion in materials: A review. Advanced Materials, 29(50), 1703546.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Muhannad F. AL- Saadony, Nasir A. Naser (Author)
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Licensed under a CC-BY license: https://creativecommons.org/licenses/by-nc-sa/4.0/
