Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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BARRIER OPTIONS UNDER THE MFBM WITH JUMPS : APPLICATION OF THE BDF2 METHOD

  • Choi, Heungsu (Department of Mathematics Chungnam National University) ;
  • Lee, Younhee (Department of Mathematics Chungnam National University)
  • Received : 2020.01.31
  • Accepted : 2020.02.06
  • Published : 2020.02.15
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Abstract

In this paper we consider a mixed fractional Brownian motion (mfBm) with jumps. The prices of European barrier options can be evaluated by solving a partial integro-differential equation (PIDE) with variable coefficients, which is derived from the mfBm with jumps. The 2-step backward differentiation formula (BDF2 method) proposed in [6] is applied with the second-order convergence rate in the time and spatial variables. Numerical simulations are carried out to observe the convergence behaviors of the BDF2 method under the mfBm with the Kou model.

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References

  1. L. V. Ballestra, G. Pacelli, and D. Radi, A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion, Chaos Solitons Fractals, 87 (2016), 240-248. https://doi.org/10.1016/j.chaos.2016.04.008
  2. T. Bjork and H. Hult, A note on Wick products and the fractional Black-Scholes model, Financ. Stoch., 9 (2005), 197-209. https://doi.org/10.1007/s00780-004-0144-5
  3. P. Cheridito, Mixed fractional Brownian motion, Bernoulli, 7 (2001), 913-934. https://doi.org/10.2307/3318626
  4. P. Cheridito, Arbitrage in fractional Brownian motion models, Financ. Stoch., 7 (2003), 533-553. https://doi.org/10.1007/s007800300101
  5. Y. d'Halluin, P. A. Forsyth, and K. R. Vetzal, Robust numerical methods for contingent claims under jump diffusion processes, IMA J. Numer. Anal., 25 (2005), 87-112. https://doi.org/10.1093/imanum/drh011
  6. S. Lee and Y. Lee, Stability of numerical methods under the regime-switching jump-diffusion model with variable coefficients, ESAIM Math. Model. Numer. Anal., 53 (2019), 1741-1762. https://doi.org/10.1051/m2an/2019035
  7. B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437. https://doi.org/10.1137/1010093
  8. R. C. Merton, Theory of rational option pricing, Bell J. Econ. Manage. Sci., 4 (1973), 141-183. https://doi.org/10.2307/3003143
  9. F. Shokrollahi and A. Kilicman, Pricing currency option in a mixed fractional Brownian motion with jumps environment, Math. Probl. Eng., 2014 (2014), Art. ID 858210, 13 pages.
  10. L. Sun, Pricing currency options in the mixed fractional Brownian motion, Physica A, 392 (2013), 441-3458.
  11. W. L. Xiao, W. G. Zhang, X. L. Zhang, and X. L. Zhang, Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm, Physica A, 391 (2012), 6418-6431. https://doi.org/10.1016/j.physa.2012.07.041