Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 537571, 17 pages
Research Article

Modeling and Pricing of Variance and Volatility Swaps for Local Semi-Markov Volatilities in Financial Engineering

1Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N 1N4
2Department of Mathematics, University of Rome ‘‘La Sapienza’’, Via del Castro, Laurenziano 9, 00161 Rome, Italy

Received 28 July 2010; Accepted 14 October 2010

Academic Editor: G. Rega

Copyright © 2010 Anatoliy Swishchuk and Raimondo Manca. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. K. Demeterfi, E. Derman, M. Kamal, and J. Zou, “A guide to volatility and variance swaps,” The Journal of Derivatives, vol. 6, no. 4, pp. 9–32, 1999. View at Google Scholar
  2. J. Hull, Options, Futures and Other Derivatives, Prentice Hall, Upper Saddle River, NJ, USA, 4th edition, 2000.
  3. A. Swishchuk, “Pricing of variance and volatility swaps with semi-Markov volatilities,” Canadian Applied Mathematics Quarterly. Accepted.
  4. F. Black and M. Scholes, “The pricing of options and corporate liabilities,” Journal of Political Economy, vol. 81, pp. 637–654, 1973. View at Google Scholar
  5. J. C. Cox, S. A. Ross, and M. Rubinstein, “Option pricing: a simplified approach,” Journal of Financial Economics, vol. 7, no. 3, pp. 229–263, 1979. View at Google Scholar · View at Scopus
  6. B. Dupire, “Pricing with a smile,” Risk, vol. 7, no. 1, pp. 18–20, 1994. View at Google Scholar
  7. E. Derman and I. Kani, “Riding on a smile,” Risk, vol. 7, no. 2, pp. 32–39, 1994. View at Google Scholar
  8. P. Wilmott, S. Howison, and J. Dewynne, Option Pricing: Mathematical Models and Computations, Oxford Financial Press, Oxford, UK, 1995.
  9. R. C. Merton, “Theory of rational option pricing,” The Rand Journal of Economics, vol. 4, pp. 141–183, 1973. View at Google Scholar
  10. J. Hull and A. White, “The pricing of options on assets with stochastic volatilities,” The Journal of Finance, vol. 42, pp. 281–300, 1987. View at Google Scholar
  11. S. Heston, “A closed-form solution for options with stochastic volatility with applications to bond and currency options,” Review of Financial Studies, vol. 6, pp. 327–343, 1993. View at Google Scholar
  12. R. J. Elliott and A. V. Swishchuk, “Pricing options and variance swaps in Markov-modulated Brownian markets,” in Hidden Markov Models in Finance, R. Mamon and R. Elliott, Eds., vol. 104 of Internat. Ser. Oper. Res. Management Sci., pp. 45–68, Springer, New York, NY, USA, 2007. View at Google Scholar
  13. A. Swishchuk, Random Evolutions and Their Applications. New Trends, vol. 504 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
  14. M. Avellaneda, A. Levy, and A. Paras, “Pricing and hedging derivative securities in markets with uncertain volatility,” Applied Mathematical Finance, vol. 2, pp. 73–88, 1995. View at Google Scholar
  15. Y. Kazmerchuk, A. Swishchuk, and J. Wu, “A continuous-time Garch model for stochastic volatility with delay,” Canadian Applied Mathematics Quarterly, vol. 13, no. 2, pp. 123–149, 2005. View at Google Scholar
  16. T. Bollerslev, “Generalized autoregressive conditional heteroskedasticity,” Journal of Econometrics, vol. 31, no. 3, pp. 307–327, 1986. View at Publisher · View at Google Scholar
  17. A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models, Theory, vol. 3 of Advanced Series on Statistical Science & Applied Probability, World Scientific, River Edge, NJ, USA, 1999. View at Publisher · View at Google Scholar
  18. David G. Hobson and L. C. G. Rogers, “Complete models with stochastic volatility,” Mathematical Finance, vol. 8, no. 1, pp. 27–48, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. A. Javaheri, P. Wilmott, and E. Haug, “GARCH and volatility swaps,” Quantitative Finance, vol. 4, no. 5, pp. 589–595, 2004. View at Publisher · View at Google Scholar
  20. O. Brockhaus and D. Long, “Volatility swaps made simple,” Risk, vol. 2, no. 1, pp. 92–96, 2000. View at Google Scholar
  21. J. Janssen, R. Manca, and E. Volpe, Mathematical Finance: Deterministic and Stochastic Models, Wiley, New York, NY, USA, 2008. View at Publisher · View at Google Scholar
  22. V. Korolyuk and A. Swishchuk, Semi-Markov Random Evolutions, vol. 308 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995.
  23. E. B. Dynkin, Markov Processes. Vols. I, II, Die Grundlehren der Mathematischen Wissenschaften, Bände 121-122, Academic Press Publishers, New York, NY, USA, 1965. View at Zentralblatt MATH
  24. P. Carr and D. Madan, “Towards a Theory of Volatility Trading,” in Volatility: New Estimation Techniques for Pricing Derivatives, pp. 417–427, Risk Book Publications, London, UK, 1998. View at Google Scholar
  25. G. D'Amico, J. Janssen, and R. Manca, “Valuing credit default swap in a non-homogeneous semi-Markovian rating based model,” Computational Economics, vol. 29, no. 2, pp. 119–138, 2007. View at Publisher · View at Google Scholar · View at Scopus