Call warrants pricing formula under mixed-fractional Brownian motion with Merton jump-diffusion

Mixed fractional Brownian motion (MFBM) is a linear combination of a Brownian motion and an independent fractional Brownian motion which may overcome the problem of arbitrage, while a jump process in time series is another problem to be address in modeling stock prices.  This study models call warrants with MFBM and includes the jump process in its dynamics.  The pricing formula for a warrant with mixed-fractional Brownian motion and jump, is obtained via quasi-conditional expectation and risk-neutral valuation.

mixed-fractional Brownian motion
Merton jump diffusion
call warrant
  1. Black F., Scholes M.  The pricing of options and corporate liabilities.  Journal of Political Economy.  81 (3), 637–654 (1973).
  2. Cont R.  Empirical properties of asset returns: stylized facts and statistical issues.  Quantitative Finance.  1 (2), 223–236 (2010).
  3. Cajueiro D. O., Tabak B. M.  Long-range dependence and multifractality in the term structure of LIBOR interest rates.  Physica A.  373, 603–614 (2007).
  4. Huang T.-C., Tu Y.-C., Chou H.-C.  Long memory and the relation between options and stock prices.  Finance Research Letters.  12, 77–91 (2015).
  5. Rogers L. C. G.  Arbitrage with fractional Brownian motion.  Mathematical Finance.  7 (1), 95–105 (1997).
  6. Cheridito P.  Arbitrage in fractional Brownian motion models.  Finance and Stochastics.  7, 533–553 (2003).
  7. Li J., Xiang K., Luo C.  Pricing study on two kinds of power options in jump-diffusion models with fractional Brownian motion and stochastic rate.  Applied Mathematics.  5 (16), 2426–2441 (2014).
  8. Wang C., Zhou S., Yang J.  The Pricing of Vulnerable Options in a Fractional Brownian Motion Environment.  Discrete Dynamics in Nature and Society.  2015, 579213 (2015).
  9. El-Nouty C.  The fractional mixed fractional Brownian motion.  Statistics & Probability Letters.  65 (2), 111–120 (2003).
  10. Mishura Y. S.  Stochastic calculus for fractional Brownian motions and related processes.  Springer Berlin, Heidelberg (2008).
  11. Kuznetsov Yu. A.  The absence of arbitrage in a model with fractal Brownian motion.  Russian Mathematical Surveys.  54 (4), 847 (1999).
  12. Björk T., Hult H.  A note on Wick products and the fractional Black–Scholes model.  Finance and Stochastics.  9, 197–209 (2005).
  13. Cheridito P.  Mixed Fractional Brownian Motion.  Bernoulli.  7 (6), 913–934 (2001).
  14. Dominique C. R., Rivera–Solis L. E. S.  Mixed fractional Brownian motion, short and long-term dependence and economic conditions: The case of the S&P-500 Index.  International Business and Management.  3 (2), 1–6 (2011).
  15. Prakasa Rao B. L. S.  Pricing geometric Asian power options under mixed fractional Brownian motion environment.  Physica A.  446, 92–99 (2016).
  16. Zhang W.-G., Li Z., Liu Y.-J.  Analytical pricing of geometric Asian power options on an underlying driven by mixed fractional Brownian motion.  Physica A.  490, 402–418 (2018).
  17. Chen W., Yan B., Lian G., Zhang Y.  Numerically pricing American options under the generalized mixed fractional Brownian motion model.  Physica A.  451, 180–189 (2016).
  18. Shokrollahi F., Kiliçman A.  Pricing currency option in a mixed fractional Brownian motion with jumps environment.  Mathematical Problems in Engineering.  2014, 858210 (2014).
  19. Shokrollahi F.  Pricing compound and extendible options under mixed fractional Brownian motion with jumps.  Axioms.  8 (2), 39 (2019).
  20. Shokrollahi F., Ahmadian D., Ballestra L.  Actuarial strategy for pricing Asian options under a mixed fractional Brownian motion with jumps.  Preprint arXiv:2105.06999 (2021).
  21. Xiao W.-L., Zhang W.-G., Zhang X., Zhang X.  Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm.  Physica A.  391 (24), 6418–6431 (2012).
  22. Merton R. C.  Option pricing when underlying stock returns are discontinuous.  Journal of Financial Economics.  3 (1–2), 125–144 (1976).
  23. Biagini F., Hu Y., Øksendal B., Zhang T.  Stochastic Calculus for Fractional Brownian Motion and Applications. Springer London (2008).
  24. Miao J., Yang X.  Pricing Model for Convertible Bonds: A Mixed Fractional Brownian Motion with Jumps.  East Asian Journal on Applied Mathematics.  5 (3), 222–237 (2015).
  25. Schulz G. U., Trautmann S.  Robustness of option-like warrant valuation.  Journal of Banking & Finance.  18 (5), 841–859 (1994).
  26. Yanishevskyi V. S., Nodzhak L. S.  The path integral method in interest rate models.  Mathematical Modeling and Computing.  8 (1),  125–136 (2021).
  27. Yanishevskyi V. S., Baranovska S. P.  Path integral method for stochastic equations of financial engineering.  Mathematical Modeling and Computing.  9 (1), 166–177 (2022).