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Forward Start Foreign Exchange Options Under Heston’s Volatility and the CIR Interest Rates

  • Rehez Ahlip
  • Marek Rutkowski

Abstract

We examine the valuation of forward start foreign exchange options in the Heston (Rev. Financ. Stud. 6:327–343, 1993) stochastic volatility model for the exchange rate combined with the CIR (see Cox et al. in Econometrica 53:385–408, 1985) dynamics for the domestic and foreign interest rates. The instantaneous volatility is correlated with the dynamics of the exchange rate, whereas the domestic and foreign short-term rates are assumed to be independent of the dynamics of the exchange rate volatility. The main results are derived using the probabilistic approach combined with the Fourier inversion technique developed in Carr and Madan (J. Comput. Finance 2:61–73, 1999). They furnish two alternative semi-analytical formulae for the price of the forward start foreign exchange European call option. As was argued in Ahlip and Rutkowski (Quant. Finance 13:955–966, 2013), the setup examined here is the only analytically tractable version of the foreign exchange market model that combines the Heston stochastic volatility model for the exchange rate with the CIR dynamics for interest rates.

Keywords

Option pricing Heston stochastic volatility model Forward start options Interest rates 

Mathematics Subject Classification (2010)

91G20 91G30 

Notes

Acknowledgements

The research of M. Rutkowski was supported under Australian Research Council’s Discovery Projects funding scheme (project number DP0881460). The paper is in the final form and no similar paper has been or is being submitted elsewhere. The authors are grateful to anonymous referees for their detailed and insightful reports.

References

  1. 1.
    Ahlip, R., Rutkowski, M.: Pricing of foreign exchange options under the Heston stochastic volatility model and the CIR interest rates. Quant. Finance 13, 955–966 (2013) CrossRefGoogle Scholar
  2. 2.
    Amerio, E.: Forward start option pricing with stochastic volatility: a general framework. In: Locke, E. (ed.) Financial Engineering and Applications: Proceedings of the Fourth IASTED International Conference, pp. 44–53. Acta Press, Calgary (2007) Google Scholar
  3. 3.
    Carr, P., Madan, D.: Option valuation using the fast Fourier transform. J. Comput. Finance 2, 61–73 (1999) Google Scholar
  4. 4.
    Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of term structure of interest rates. Econometrica 53, 385–408 (1985) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Grzelak, L.A., Oosterlee, C.W.: On the Heston model with stochastic interest rates. SIAM J. Financ. Math. 2, 255–286 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grzelak, L.A., Oosterlee, C.W.: On cross-currency models with stochastic volatility and correlated interest rates. Appl. Math. Finance 19, 1–35 (2012) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Grzelak, L.A., Oosterlee, C.W., Van Weeren, S.: Extension of stochastic volatility equity models with the Hull-White interest rate process. Quant. Finance 12, 89–105 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6, 327–343 (1993) CrossRefGoogle Scholar
  10. 10.
    Jeanblanc, M., Yor, M., Chesney, M.: Mathematical Methods for Financial Markets. Springer, Berlin (2009) CrossRefzbMATHGoogle Scholar
  11. 11.
    Kruse, S., Nögel, U.: On the pricing of forward starting options in Heston’s model on stochastic volatility. Finance Stoch. 9, 233–250 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lipton, A.: Mathematical Methods for Foreign Exchange Options: A Financial Engineer’s Approach, pp. 608–611. World Scientific, New Jersey (2001) CrossRefGoogle Scholar
  13. 13.
    Lucic, V.: Forward start options in stochastic volatility models. In: Wilmott, P. (ed.) The Best of Wilmott 1, pp. 413–420. John Wiley, Chichester (2004) Google Scholar
  14. 14.
    Musiela, M., Rutkowski, M.: Martingale Methods in Financial Modelling, 2nd edn. Springer, Berlin (2005) zbMATHGoogle Scholar
  15. 15.
    Schöbel, R., Zhu, J.: Stochastic volatility with an Ornstein-Uhlenbeck process: an extension. Eur. Finance Rev. 3, 23–46 (1999) CrossRefzbMATHGoogle Scholar
  16. 16.
    Van Haastrecht, A., Pelsser, A.: Generic pricing of foreign exchange, inflation and stock options under stochastic interest rates and stochastic volatility. Quant. Finance 11, 665–691 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Van Haastrecht, A., Lord, R., Pelsser, A., Schrager, D.: Pricing long-maturity equity and FX derivatives with stochastic interest rates and stochastic volatility. Insur. Math. Econ. 45, 436–448 (2009) CrossRefzbMATHGoogle Scholar
  18. 18.
    Vasicek, O.: An equilibrium characterisation of the term structure. J. Financ. Econ. 5, 177–188 (1977) CrossRefGoogle Scholar
  19. 19.
    Windcliff, H.A., Forsyth, P.A., Vetzal, K.R.: Numerical methods and volatility models for valuing cliquet options. Appl. Math. Finance 13, 353–386 (2006) CrossRefzbMATHGoogle Scholar
  20. 20.
    Wong, B., Heyde, C.C.: On the martingale property of stochastic exponentials. J. Appl. Probab. 41, 654–664 (2004) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.School of Computing and MathematicsUniversity of Western SydneyPenrith SouthAustralia
  2. 2.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

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