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Journal of Applied Mathematics and Stochastic Analysis
Volume 2009, Article ID 215817, 16 pages
http://dx.doi.org/10.1155/2009/215817
Research Article

On Modelling Long Term Stock Returns with Ergodic Diffusion Processes: Arbitrage and Arbitrage-Free Specifications

School of Actuarial Studies, Australian School of Business, University of New South Wales, NSW 2052, Australia

Received 13 March 2009; Accepted 29 July 2009

Academic Editor: Vo Anh

Copyright © 2009 Bernard Wong. 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. R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004. View at MathSciNet
  2. J.-P. Fouque, G. Papanicolaou, and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000. View at MathSciNet
  3. M. Sherris, “The valuation of option features in retirement benefits,” The Journal of Risk and Insurance, vol. 62, no. 3, pp. 509–534, 1995. View at Google Scholar
  4. D. Bauer, R. Kiesel, A. Kling, and J. Ruß, “Risk-neutral valuation of participating life insurance contracts,” Insurance: Mathematics & Economics, vol. 39, no. 2, pp. 171–183, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. A. Milevsky and T. S. Salisbury, “Financial valuation of guaranteed minimum withdrawal benefits,” Insurance: Mathematics & Economics, vol. 38, no. 1, pp. 21–38, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. K. Zaglauer and D. Bauer, “Risk-neutral valuation of participating life insurance contracts in a stochastic interest rate environment,” Insurance: Mathematics & Economics, vol. 43, no. 1, pp. 29–40, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. Cairns, “Some notes on the dynamics and optimal control of stochastic pension fund models in continuous time,” ASTIN Bulletin, vol. 30, no. 1, pp. 19–55, 2000. View at Google Scholar
  8. H. U. Gerber and E. S. W. Shiu, “Geometric Brownian motion models for assets and liabilities: from pension funding to optimal dividends,” North American Actuarial Journal, vol. 7, no. 3, pp. 37–56, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. Z. Stamos, “Optimal consumption and portfolio choice for pooled annuity funds,” Insurance: Mathematics & Economics, vol. 43, no. 1, pp. 56–68, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. Hardy, Investment Guarantees: Modeling and Risk Management for Equity-Linked Life Insurance, John Wiley & Sons, River Edge, NJ, USA, 2003.
  11. H. Schmidli, Stochastic Control in Insurance, Probability and Its Applications, Springer, London, UK, 2008. View at MathSciNet
  12. M. A. Milevsky, The Calculus of Retirement Income: Financial Models for Pension Annuities and Life Insurance, Cambridge University Press, Cambridge, UK, 2006. View at MathSciNet
  13. T. Møller and M. Steffensen, Market-Valuation Methods in Life and Pension Insurance, Cambridge University Press, Cambridge, UK, 2007.
  14. B. Bibby and M. Sørensen, “A hyperbolic diffusion model for stock prices,” Finance and Stochastics, vol. 1, pp. 25–41, 1997. View at Google Scholar
  15. T. Rydberg, “A note on the existence of equivalent martingale measures in a Markovian setting,” Finance and Stochastics, vol. 1, pp. 251–257, 1997. View at Google Scholar
  16. T. Rydberg, “Generalized hyperbolic diffusion processes with applications in finance,” Mathematical Finance, vol. 9, no. 2, pp. 183–201, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. M. Harrison and D. M. Kreps, “Martingales and arbitrage in multiperiod securities markets,” Journal of Economic Theory, vol. 20, no. 3, pp. 381–408, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. J. M. Harrison and S. R. Pliska, “Martingales and stochastic integrals in the theory of continuous trading,” Stochastic Processes and Their Applications, vol. 11, no. 3, pp. 215–260, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. F. Delbaen and W. Schachermayer, “A general version of the fundamental theorem of asset pricing,” Mathematische Annalen, vol. 300, no. 3, pp. 463–520, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. H. J. Engelbert and W. Schmidt, “On solutions of one-dimensional stochastic differential equations without drift,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 68, no. 3, pp. 287–314, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, New York, NY, USA, 1981. View at MathSciNet
  22. T. Conley, L. Hansen, E. Luttmer, and J. Scheinkman, “Short-term interest rates as subordinated diffusion,” Review of Financial Studies, vol. 10, pp. 525–577, 1997. View at Google Scholar
  23. J. Nicolau, “Processes with volatility-induced stationarity: an application for interest rates,” Statistica Neerlandica, vol. 59, no. 4, pp. 376–396, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. B. Bibby, I. M. Skovgaard, and M. Sørensen, “Diffusion-type models with given marginal distribution and autocorrelation function,” Bernoulli, vol. 11, no. 2, pp. 191–220, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. Borkovec and C. Klüppelberg, “Extremal behavior of diffusion models in finance,” Extremes, vol. 1, no. 1, pp. 47–80, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. B. Dupire, “Pricing with a smile,” Risk, vol. 7, pp. 18–20, 1994. View at Google Scholar
  27. D. B. Madan and M. Yor, “Making Markov martingales meet marginals: with explicit constructions,” Bernoulli, vol. 8, no. 4, pp. 509–536, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. P. Dybvig and C. Huang, “Nonnegative wealth, absence of arbitrage, and feasible consumption plans,” Review of Financial Studies, vol. 1, pp. 377–401, 1988. View at Google Scholar
  29. N. Kazamaki, “Continuous exponential martingales and BMO,” in Lecture Notes in Mathematics, vol. 1579, Springer, Berlin, Germany, 1994. View at Google Scholar
  30. R. Liptser and A. Shiryaev, Statistics of Random Processes, vol. 1, Springer, Berlin, Germany, 1977.
  31. T. T. Kadota and L. A. Shepp, “Conditions for absolute continuity between a certain pair of probability measures,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 16, pp. 250–260, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. F. Delbaen and H. Shirakawa, “No arbitrage condition for positive diffusion price processes,” Asia-Pacific Financial Markets, vol. 9, pp. 159–168, 2002. View at Google Scholar
  33. B. Wong and C. C. Heyde, “On the martingale property of stochastic exponentials,” Journal of Applied Probability, vol. 41, no. 3, pp. 654–664, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. L. C. G. Rogers and L. A. M. Veraart, “A stochastic volatility alternative to SABR,” Journal of Applied Probability, vol. 45, no. 4, pp. 1071–1085, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. S. Levental and A. V. Skorohod, “A necessary and sufficient condition for absence of arbitrage with tame portfolios,” The Annals of Applied Probability, vol. 5, no. 4, pp. 906–925, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. D. Heath, R. Jarrow, and A. Morton, “Bond pricing and the term structure of interest rates: a new methodology for contingent claim valuation,” Econometrica, vol. 60, pp. 77–105, 1992. View at Google Scholar
  37. M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling, vol. 36 of Stochastic Modelling and Applied Probability, Springer, Berlin, Germany, 2nd edition, 2005. View at MathSciNet