Hyperbolic Symmetrization of Heston Type Diffusion

  • Yuuki IdaEmail author
  • Tsuyoshi Kinoshita


The symmetrization of diffusion processes was originally introduced by Imamura, Ishigaki and Okumura, and was applied to pricing of barrier options. The authors of the present paper previously introduced in Ida et al. (Pac J Math Ind 10:1, 2018) a hyperbolic version of the symmetrization of a diffusion by symmetrizing drift coefficient in view of applications under a SABR model which is transformed to a hyperbolic Brownian motion with drift. In the present paper, in order to apply the hyperbolic symmetrization technique to Heston model, we introduce an extension where diffusion coefficient is also symmetrized. Some numerical results are also presented.


Heston model Symmetrization Euler–Maruyama scheme Hyperbolic space Numerical simulation 



The authors deeply thank Prof. Jirô Akahori for his warm support.


  1. Akahori, J., & Imamura, Y. (2014). On a symmetrization of diffusion processes. Quantitative Finance, 14(7), 1211–1216.CrossRefGoogle Scholar
  2. Carr, P., & Lee, R. (2009). Put-call symmetry: Extensions and applications. Mathematical Finance, 19(4), 523–560.CrossRefGoogle Scholar
  3. Ida, Y., Kinoshita, T., & Matsumoto, T. (2018). Symmetrization associated with hyperbolic reflection principle. Pacific Journal of Mathematics for Industry, 10, 1.CrossRefGoogle Scholar
  4. Imamura, Y., Ishigaki, Y., & Okumura, T. (2014). A numerical scheme based on semi-static hedging strategy. Monte Carlo Methods and Applications, 20(4), 223–235.CrossRefGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesRitsumeikan UniversityKusatsuJapan
  2. 2.Graduate School of Science and EngineeringRitsumeikan UniversityKusatsuJapan

Personalised recommendations