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Decisions in Economics and Finance

, Volume 37, Issue 2, pp 319–327 | Cite as

Symmetry and Bates’ rule in Ornstein–Uhlenbeck stochastic volatility models

  • José Fajardo
Article

Abstract

We find necessary and sufficient conditions for the market symmetry property, introduced by Fajardo and Mordecki (Quant Finance 6(3):219–227, 2006), to hold in the Ornstein–Uhlenbeck stochastic volatility model, henceforth OU–SV. In particular, we address the non-Gaussian OU–SV model proposed by Barndorff-Nielsen and Shephard (J R Stat Soc B 63(Part 2):167–241, 2001). Also, we prove the Bates’ rule for these models.

Keywords

Barndorff-Nielsen and Shephard Model Symmetry Bates’s rule Ornstein–Uhlenbeck process 

JEL Classification

C52 G10 

References

  1. Barndorff-Nielsen, O.E., Shephard, N.: Non-Gaussian Orsntein–Uhlenbeck-Based Models and some of their uses on Financial Economics. J. R. Stat. Soc. B 63(Part 2), 167–241 (2001)CrossRefGoogle Scholar
  2. Bates, D.: The crash of ’87—was it expected? The evidence from options markets. J. Finance 46(3), 1009–1044 (1991)CrossRefGoogle Scholar
  3. Bates, D.: Dollar jump fears, 1984–1992: distributional abnormalities implicit in foreign currency futures options. J. Int. Money Finance 15(1), 65–93 (1996)CrossRefGoogle Scholar
  4. Bates, D.: The skewness premium: option pricing under asymmetric processes. Adv. Futures Options Res. 9, 51–82 (1997)Google Scholar
  5. Benth, F.: The stochastic volatility model of Barndorff-Nielsen and Shephard in commodity markets. Math. Finance 21(4), 595–625 (2011)Google Scholar
  6. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Political Econ 81, 637–659 (1973)CrossRefGoogle Scholar
  7. Carr, P., Lee, R.: Put call symmetry: extensions and applications. Math. Finance 19(4), 523–560 (2009)CrossRefGoogle Scholar
  8. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. CRC Financial Mathematics Series. Chapman& Hall, London (2004)Google Scholar
  9. Eberlein, E., Papapantoleon, A., Shiryaev, A.N.: On the duality principle in option pricing: semimartingale setting. Finance Stochast. 12(2), 265–292 (2008)CrossRefGoogle Scholar
  10. Fajardo, J., Mordecki, E.: Symmetry and duality in Lèvy markets. Quant. Finance 6(3), 219–227 (2006)CrossRefGoogle Scholar
  11. Fajardo, J., Mordecki, E.: Market symmetry in time changed brownian models. Finance Res. Lett. 7(1), 53–59 (2010)CrossRefGoogle Scholar
  12. Fajardo, J., Mordecki, E.: Skewness premium with Lévy processes. Quant. Finance (2011). doi: 10.1080/14697688.2011.618809
  13. Heston, S.: Closed-form solution for options with stochastic volatiliy, with application to bond and currency options. Rev. Financ. Stud. 6, 327–343 (1993)CrossRefGoogle Scholar
  14. Hubalek, F., Posedel, P.: Joint analysis and estimation of stock prices and trading volume in Barndorff-Nielsen and Shephard stochastic volatility models. Quant. Finance 11(6), 917–993 (2011)CrossRefGoogle Scholar
  15. Hubalek, F., Sgarra, C.: On the esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Sheppard stochastic volatility models with jumps. Stochast. Process. Appl. 119(7), 2137–2157 (2009)CrossRefGoogle Scholar
  16. Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003)CrossRefGoogle Scholar
  17. Kallsen, J., Pauwels, A.: Variance-optimal hedging in general affine stochastic volatility models. Adv. Appl. Probab. 42, 83–105 (2010)CrossRefGoogle Scholar
  18. Kallsen, J., Shiryaev, A.N.: The cumulant process and Esscher’s change of measure. Finance Stochast. 6, 397–428 (2002)CrossRefGoogle Scholar
  19. Merton, R.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141–183 (1973)CrossRefGoogle Scholar
  20. Molchanov, I., Schmutz, M.: Multivariate extension of put-call symmetry. SIAM J. Financ. Math. 1, 396–426 (2010)CrossRefGoogle Scholar
  21. Nicolato, E., Vernardos, E.: Option pricing in stochastic volatility models of the Ornstein–Uhlembeck type. Math. Finance 13(4), 445–466 (2003)CrossRefGoogle Scholar
  22. Schmutz, M.: Semi-static hedging for certain Margrabe-type options with barriers. Quant. Finance 11(7), 979–986 (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Brazilian School of Public and Business AdministrationGetulio Vargas FoundationRio de JaneiroBrazil

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