Fractional Brownian motion in financial engineering models

: pp. 445–457
Received: January 10, 2023
Revised: May 12, 2023
Accepted: May 18, 2023
Lviv Polytechnic National University
Lviv Polytechnic National University

An application of fractional Brownian motion (fBm) is considered in stochastic financial engineering models.  For the known Fokker–Planck equation for the fBm case, a solution for transition probability density for the path integral method was built.  It is shown that the mentioned solution does not result from the Gaussian unit of fBm with precise covariance.  An expression for approximation of fBm covariance was found for which solutions are found based on the Gaussian measure of fBm and those found based on the known Fokker–Planck equation match.

  1. Lyuu Y.-D.  Financial Engineering and Computation: Principles, Mathematics, and Algorithms.  Cambridge University Press (2004).
  2. Gardiner C.  Handbook of Stochastic Methods for Springer Berlin, Heidelberg.  Springer Berlin, Heidelberg (2004).
  3. Paul W., Baschnagel J.  Stochastic Processes. From Physics to Finance.  Springer Cham (2013).
  4. Øksendal B.  Stochastic Differential Equations: An Introduction with Applications.  Springer Berlin, Heidelberg (2003).
  5. Sawal A. S., Ibrahim S. N., Laham M. F.  The valuation of knock-out power calls under Black–Scholes framework.  Mathematical Modeling and Computing.  9 (1), 57–64 (2022).
  6. Sawal A. S., Ibrahim S. N. I., Roslan T. R. N.  Pricing equity warrants with jumps, stochastic volatility, andstochastic interest rates.  Mathematical Modeling and Computing.  9 (4), 882–891 (2022).
  7. Yanishevskyi V. S., Nodzhak L. S.  The path integral method in interest rate models.  Mathematical Modeling and Computing.  8 (1), 125–136 (2021).
  8. Hassler U.  Stochastic Processes and Calculus: An Elementary Introduction with Applications.  Springer Cham (2016).
  9. Umarov S., Hahn M., Kobayashi K.  Beyond the triangle: Brownian motion, Ito calculus, and Fokker–Planck equation — Fractional Generalizations.  World Scientific Publishing Co. Pte. Ltd. (2018).
  10. Nourdin I.  Selected Aspects of Fractional Brownian Motion.  Springer Milano (2012).
  11. Bender C.  An Ito formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter.  Stochastic Processes and their Applications.  104 (1), 81–106 (2003).
  12. Nualart D.  The Malliavin Calculus and Related Topics.  Springer Berlin, Heidelberg (2006).
  13. Ahmadian D., Ballestra L. V.  Pricing geometric Asian rainbow options under the mixed fractional Brownian motion.  Physica A: Statistical Mechanics and its Applications.  555, 124458 (2020).
  14. Araneda A. A.  The fractional and mixed-fractional CEV model.  Journal of Computational and Applied Mathematics.  363, 106–123 (2020).
  15. Araneda A. A., Bertschinger N.  The sub-fractional CEV model.  Physica A: Statistical Mechanics and its Applications.  573, 125974 (2021).
  16. Ibrahim S. N. I., Laham M. F.  Call warrants pricing formula under mixed-fractional Brownian motion with Merton jump-diffusion.  Mathematical Modeling and Computing.  9 (4), 892–897 (2022).
  17. Herzog B.  Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion.  Mathematics.  10 (3), 340 (2022).
  18. Araneda A. A.  European option pricing under generalized fractional Brownian motion.  ArXiv:2108.12042v1 (2021).
  19. Yanishevskyi V. S., Baranovska S. P.  Path integral method for stochastic equations of financial engineering.  Mathematical Modeling and Computing.  9 (1), 166–177 (2022).
  20. Osu B. O., Ifeoma C. A.  Fractional Black Scholes Option Pricing with Stochastic Arbitrage Return.  International Journal of Partial Differential Equations and Applications.  4 (2), 20–24 (2016).
  21. Calvo I., Sánchez R., Carreras B. A.  Fractional Lévy motion through path integrals.  Journal of Physics A: Mathematical and Theoretical.  42 (5), 055003 (2009).
  22. Mishura Y., Zili M.  Stochastic Analysis of Mixed Fractional Gaussian Processes.  ISTE Press Ltd, London, and Elsevier Ltd, Oxford (2018).
  23. Yan L., Shen G., He K.  Itô's formula for a sub-fractional Brownian motion.  Communications on Stochastic Analysis.  5 (1), 135–159 (2011).
  24. Chaichian M., Demichev A.  Path integrals in physics. Stochastic processes and quantum mechanics.  CRC Press (2001).
  25. Goovaertsa M., Schepper A. D., Decampsa M.  Closed-form approximations for diffusion densities: an integral path approach.  Journal of Computational and Applied Mathematics.  164165, 337–364 (2004).
  26. Kleinert H.  Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets.  World Scientific (2004).
  27. Ascione G., Mishura Y., Pirozzi E.  Fractional Ornstein-Uhlenbeck Process with Stochastic Forcing, and its Applications.  Methodology and Computing in Applied Probability.  23, 53–84 (2021).
Mathematical Modeling and Computing, Vol. 10, No. 2, pp. 445–457 (2023)