Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

PRICING EXTERNAL-CHAINED BARRIER OPTIONS WITH EXPONENTIAL BARRIERS

  • Jeon, Junkee (Department of Mathematical Sciences Seoul National University) ;
  • Yoon, Ji-Hun (Department of Mathematics Pusan National University)
  • Received : 2015.09.26
  • Published : 2016.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

External barrier options are two-asset options with stochastic variables where the payoff depends on one underlying asset and the barrier depends on another state variable. The barrier state variable determines whether the option is knocked in or out when the value of the variable is above or below some prescribed barrier level. This paper derives the explicit analytic solution of the chained option with an external single or double barrier by utilizing the probabilistic methods - the reflection principle and the change of measure. Before we do this, we examine the closed-form solution of the external barrier option with a single or double-curved barrier using the methods of image and double Mellin transforms. The exact solution of the external barrier option price enables us to obtain the pricing formula of the chained option with the external barrier more easily.

Keywords

References

  1. P. Buchen, Image Options and the road to barriers, Risk Magazine 14 (2001), no. 9, 127-130.
  2. P. Buchen and O. Konstandatos, A new approach to pricing double-Barrier options with arbitrary payoffs and exponential boundaries, Appl. Math. Finance 16 (2009), no. 6, 497-515. https://doi.org/10.1080/13504860903075480
  3. P. Carr, Two extensions to barrier option valuation, Appl. Math. Finance 2 (1995), 173-209. https://doi.org/10.1080/13504869500000010
  4. A. Chen, E. Petrou, and M. Suchanecki, Rainbow over Paris, Technical Report, Preprint No. 516, University of Bonn, 2012.
  5. E. Hassan and K. Adem, A note on Mellin transform and partial differential equations, International J. Pure Appl. Math. 34 (2007), no. 4, 457-467.
  6. R. C. Heynen and H. M. Kat, Partial barrier options, J. Financ. Engrg. 3 (1994), 253-274.
  7. J. Jeon, J. H. Yoon, and M. Kang, Pricing vulnerable path-dependent options using integral transforms, submitted, 2015.
  8. D. Jun and H. Ku, Cross a barrier to reach barrier options, J. Math. Anal. Appl. 389 (2012), no. 2, 968-978. https://doi.org/10.1016/j.jmaa.2011.12.038
  9. D. Jun and H. Ku, Pricing chanined options with curved barriers, Math. Finance 23 (2013), 963-776.
  10. J. Kim, J. Kim, H. J. Yoo, and B. Kim, Pricing external barrier options in a regime-switching model, J. Econom. Dynam. Control 53 (2015), 123-143. https://doi.org/10.1016/j.jedc.2015.02.007
  11. N. Kunitomo and M. Ikeda, Pricing options with curved boundaries, Math. Finance 2 (1992), 275-298. https://doi.org/10.1111/j.1467-9965.1992.tb00033.x
  12. Y. K. Kwok, L. Wu, and H. Yu, Pricing multi-asset options with an external barrier, Int. J. Theor. Appl. Finance 1 (1998), 523-541. https://doi.org/10.1142/S021902499800028X
  13. R. C. Merton, Theory of rational option pricing, Bell J. Econ. Management Sci. 4 (1973), 141-183. https://doi.org/10.2307/3003143
  14. B. Oksendal, Stochastic Differential Equations, Springer, New York, 2003.
  15. E. Reiner and M. Rubinstein, Breaking down the barriers, Risk 4 (1991), 28-35.
  16. H. Y. Wong and Y. K. Kwok, Multi-asset barrier options and occupation time derivatives, Appl. Math. Finance 10 (2003), 245-266. https://doi.org/10.1080/1350486032000107352
  17. J. H. Yoon, Mellin transform method for European option pricing with Hull-White stochastic interest rate, J. Appl. Math. (2014), Article ID 759562, 7 page.
  18. J. H. Yoon and J. H. Kim, The pricing of vulnerable options with double Mellin transforms, J. Math. Anal. Appl. 422 (2015), no. 2, 838-857. https://doi.org/10.1016/j.jmaa.2014.09.015