Annals of Finance

, Volume 10, Issue 2, pp 315–332 | Cite as

Pricing and managing risks of ruin contingent life annuities under regime switching variance gamma process

Research Article

Abstract

We propose a model for valuing ruin contingent life annuities under the regime-switching variance gamma process. The Esscher transform is employed to determine the equivalent martingale measure. The PIDE approach is adopted for the pricing formulation. Due to the path dependency of the payoff of the insurance product and the non-existence of a closed-form solution for the PIDE, the finite difference method is utilized to numerically calculate the value of the product. To highlight some practical features of the product, we present a numerical example. Finally, we examine numerically the performance of a simple hedging strategy by investigating the terminal distribution of hedging errors and the associated risk measures such as the value at risk and the expected shortfall. The impacts of the frequency of re-balancing the hedging portfolio on the quality of hedging are also discussed.

Keywords

Ruin contingent life annuity Regime switching variance gamma Esscher transform Pricing and risk management 

JEL Classification

G13 G22 D52 

References

  1. Broadie, M., Glasserman, P.: Estimating security price derivatives using simulation. Manag Sci 42(2), 269–285 (1996)CrossRefGoogle Scholar
  2. Buffington, J., Elliot, R.J.: American options with regime switching. Int J Theor Appl Financ 5, 497–514 (2002)CrossRefGoogle Scholar
  3. Cont, R., Tankov, P.: Financial Modelling with Jump Process. Boca Raton, FL: Chapman & Hall/CRC Press (2004)Google Scholar
  4. Davis, M.H.A.: Option pricing in incomplete market. In: Dempster, M.A.H., Pliska, S.R. (eds.) Mathematics of Derivative Securities. Cambridge: Cambridge University Press (1997)Google Scholar
  5. Davis, M.H.A., Johansson, M.P.: Malliavin Monte Carlo Greeks for jump diffusions. Stoch Process Appl 116, 101–129 (2006)CrossRefGoogle Scholar
  6. Eberlein, U.K., Prause, K.: New insights into smile, mispricing and value at risk. J Bus 71, 371–406 (1998)CrossRefGoogle Scholar
  7. Elliott, R., Siu, T.K.: Option pricing and filtering with hidden Markov-modulated pure-jump process. Appl Math Financ 25, 1–25 (2013)CrossRefGoogle Scholar
  8. Elliott, R.J., Aggoun, L., Moore, J.B.: Hidden Markov Models: Estimation and Control, Vol. 29 of Applications of Mathematics. Berlin: Springer (1995)Google Scholar
  9. Elliott, R.J., Chan, L., Siu, T.K.: Option pricing and Esscher transform under regime switching. Ann Financ 4, 423–432 (2005)CrossRefGoogle Scholar
  10. Fard, F.A., Siu, T.K.: Pricing and managing risks of european-style options in a markovian regime-switching binomial model. Ann Finance (2012). doi:10.1007/s10436-012-0192-3:1-18
  11. Föllmer, H., Schweizer, M.: Hedging of contingent claims under incomplete information. In: Davis, M.H.A., Elliott, R.J. (eds.) Applied Stochastic Analysis, Stochastics Monographs, vol. 5, pp. 389–414. Gordon and Breach, London (1991)Google Scholar
  12. Follmer, H., Sondermann, D.: Hedging of contingent claims under incomplete information. In: Hildenbrand, W., Mas-Colell A. (eds.) Contrib. to Math. Econ., pp. 205–223 (1986)Google Scholar
  13. Fournie, E., Lasry, J.M., Lebuchoux, J., Lions, P.L.: Applications of Malliavin calculus to Monte Carlo methods in finance. Financ Stoch 5(2), 201–236 (2001)CrossRefGoogle Scholar
  14. Geman, H., Madan, D., Marc, Y.: Time changes for Levy processes. Math Financ 11, 79–96 (2001)CrossRefGoogle Scholar
  15. Gerber, H.U., Shiu, E.S.W.: Option pricing by Esscher transforms (with discussions). Trans Soc Actuar 46, 99–191 (1994)Google Scholar
  16. Goldfield, S.M., Quandt, R.E.: A Markov model for switching regressions. J Econ 1, 3–16 (1973)CrossRefGoogle Scholar
  17. Guo, X.: An explicit solution to an optimal stopping problem with regime switching. J Appl Prob 38(2), 464–481 (2001)CrossRefGoogle Scholar
  18. Hamilton, J.D.: A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384 (1989)CrossRefGoogle Scholar
  19. Hardy, M.R.: A regime-switching model of long-term stock returns. North Am Actuar J 5, 41–45 (2001)CrossRefGoogle Scholar
  20. Huang, H., Milevsky, M.A., Salisbury, T.S.: A different perspective on retirement income sustainability: introducing the ruin-contingent life annuity. J Wealth Manag 46, 89–97 (2009)CrossRefGoogle Scholar
  21. Huang, H., Milevsky, M.A., Salisbury, T.S.: Valuation and hedging of the ruin-contingent life annuity (rcla). Working paper (2012)Google Scholar
  22. James, L.F.: Poisson process partition calculus with applications to exchangeable models and bayesian nonparametrics. arXiv:math/0205093v1 [math.PR] (2002)Google Scholar
  23. James, L.F.: Bayesian Poisson process partition calculus with an application to Bayesian Levy moving averages. Ann Stat 33, 1771–1799 (2005)CrossRefGoogle Scholar
  24. Madan, D.B., Carr, P.P., Chang, E.C.: The variance gamma process and option pricing. Eur Financ Rev 2, 79–105 (1998)CrossRefGoogle Scholar
  25. Naik, V.: Option valuation and hedging strategies with jumps in the volatility of asset returns. J Financ 48(5), 1964–1984 (1993)CrossRefGoogle Scholar
  26. Quandt, R.E.: The estimation of parameters of linear regression system obeying two separate regime. J Am Stat Assoc 55, 873–880 (1958)CrossRefGoogle Scholar
  27. Rong, N., Fard, F.A.: Pricing ruin-contingent life annuity under stochastic volatility. J Risk Financ 14, 35–48 (2013)CrossRefGoogle Scholar
  28. Sato, K.I.: Levy Process and Infinitely Divisible Distributions. Cambridge: Cambridge University Press (2004)Google Scholar
  29. Schweize, M.: On minimal martingale measure and and Follmer-Schweizer decomposition. Stoch Anal Appl 13, 573–599 (1995)CrossRefGoogle Scholar
  30. Shimazaki, H., Shinomoto, S.: Kernel bandwidth optimization in spike rate estimation. J Comput Neurosci 29, 171–182 (2010)CrossRefGoogle Scholar
  31. Siu, T.K.: Fair valuation of participating policies with surrender options and regime switching. Insur Math Econ 37, 533–552 (2005)CrossRefGoogle Scholar
  32. Siu, T.K., Lau, J.W., Yang, H.: Pricing participating products under a generalized jump-diffusion model. J Appl Math Stoch Anal (2008). doi:10.1155/2008/474623
  33. Tong, H.: Threshold Models in Nonlinear Time Series Analysis. Berlin: Springer (1983)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Applied Finance and Actuarial Studies, Faculty of Business and EconomicsMacquarie UniversitySydneyAustralia

Personalised recommendations