Methodology and Computing in Applied Probability

, Volume 18, Issue 3, pp 829–845 | Cite as

Option Pricing Under Jump-Diffusion Processes with Regime Switching



We study an incomplete market model, based on jump-diffusion processes with parameters that are switched at random times. The set of equivalent martingale measures is determined. An analogue of the fundamental equation for the option price is derived. In the case of the two-state hidden Markov process we obtain explicit formulae for the option prices. Furthermore, we numerically compare the results corresponding to different equivalent martingale measures.


Jump-telegraph process Jump-diffusion process Martingales Relative entropy Financial modelling Option pricing Esscher transform 

Mathematics Subject Classification (2010)

91B28 60G44 60J75 60K99 


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  1. Bellamy N, Jeanblanc M (2000) Incompleteness of markets driven by a mixed diffusion. Finance Stochast 4:209–222MathSciNetCrossRefMATHGoogle Scholar
  2. Delbaen F, Schachermayer W (1994) General version of the fundamental theorem of asset pricing. Mathematische Annalen 300:463–520MathSciNetCrossRefMATHGoogle Scholar
  3. Di Crescenzo A, Martinucci B, Zacks S (2014) On the geometric Brownian motion with alternating trend. In: Perna C, Sibillo M (eds) Mathematical and Statistical Methods for Actuarial Sciences and Finance. Springer, pp 81–85Google Scholar
  4. Di Crescenzo A, Ratanov N (2015) On jump-diffusion processes with regime switching: martingale approach. To appear in ALEA: Latin American Journal of Probability and Mathematical StatisticsGoogle Scholar
  5. Dybvig Ph H, Ross S A (2008) arbitrage. In: Durlauf S N, Blume L E (eds) The New Palgrave Dictionary of Economics. 2nd edn. Palgrave MacmillanGoogle Scholar
  6. Elliott R J, Chan L, Siu T K (2005) Option pricing and Esscher transform under regime switching. Ann Finance 1:423–432CrossRefMATHGoogle Scholar
  7. Frittelli M (2000) The minimal entropy martingale measure and the valuation problem in incomplete markets. Math Finance 10:215–225MathSciNetCrossRefMATHGoogle Scholar
  8. Fujiwara T, Miyahara Y (2003) The minimal entropy martingale measure for geometric Lévy processes. Finance Stochast 27:509–531MathSciNetCrossRefMATHGoogle Scholar
  9. Jacobsen M (2006) Point process theory and applications. Marked point and piecewise deterministic processes. Birkhäuser, BerlinMATHGoogle Scholar
  10. Jeanblanc M, Yor M, Chesney M (2009) Mathematical methods for financial markets. Springer, HeidelbergCrossRefMATHGoogle Scholar
  11. Kolesnik A D, Ratanov N (2013) Telegraph processes and option pricing. Springer, HeidelbergCrossRefMATHGoogle Scholar
  12. Melnikov A V, Ratanov N E (2007) Nonhomogeneous telegraph processes and their application to financial market modeling. Doklady Math 75:115–117MathSciNetCrossRefMATHGoogle Scholar
  13. Ratanov N (2007) A jump telegraph model for option pricing. Quant Finan 7:575–583MathSciNetCrossRefMATHGoogle Scholar
  14. Ratanov N (2010) Option pricing model based on a Markov-modulated diffusion with jumps. Braz J Probab Stat 24:413–431MathSciNetCrossRefMATHGoogle Scholar
  15. Ratanov N (2015) Telegraph processes with random jumps and complete market models. Methodol Comput Appl Prob 17:677–695.  10.1007/s11009-013-9388-x MathSciNetCrossRefMATHGoogle Scholar
  16. Runggaldier W J (2003) Jump diffusion models. In : Handbook of Heavy Tailed Distributions in Finance (S.T. Rachev, ed.), Handbooks in Finance, Book 1 (W.Ziemba Series Ed.), Elesevier/North-Holland, pp.169-209Google Scholar
  17. Weiss G H (1994) Aspects and applications of the random walk. North-Holland, AmsterdamMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Universidad del RosarioBogotáColombia

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