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Pricing a guaranteed annuity option under correlated and regime-switching risk factors

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Abstract

A Markov-modulated affine framework for dependent risk factors is proposed to value a guaranteed annuity option (GAO). Concentrating on the important effect of volatilities, both diffusion components of the interest and mortality rates are driven by a finite-state continuous time Markov chain. We derive an explicit solution to the price of a pure endowment by solving a system of linear ordinary differential equations with the aid of the forward measure. Utilising the endowment-risk-adjusted measure with pure endowment as the corresponding numéraire, we provide an efficient and accurate formula in obtaining the GAO price. Such valuation efficiency and accuracy were demonstrated through numerical experiments. We benchmark our results with those of the Monte-Carlo simulation method and show significant differences in standard errors and computing times.

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References

  1. Ballotta L, Haberman S (2003) Valuation of guaranteed annuity conversion options. Insur: Math Econ 33:87–108

    MathSciNet  MATH  Google Scholar 

  2. Biffis E, Denuit M, Devolder P (2010) Stochatic mortality under measure changes. Scand Actuar J 4:284–311

    Article  MathSciNet  Google Scholar 

  3. Ballotta L, Haberman S (2006) The fair valuation problem of guaranteed annuity options: the stochastic mortality environment case. Insur: Math Econ 38:195–214

    MathSciNet  MATH  Google Scholar 

  4. Boyle P, Hardy M (2003) Guaranteed annuity options. ASTIN Bull 33:125–152

    Article  MathSciNet  MATH  Google Scholar 

  5. Cairns A, Blake D, Dowd K (2006) Pricing death: frameworks for the valuation and securitization of mortality risk. ASTIN Bull 36:79–120

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen L (1996) Stochastic mean and stochastic volatility: a three-factor model of the term structure of interest rates and its application to the pricing of interest rate derivatives. Financ Markets, Inst, Instrum 5:1–88

    MATH  Google Scholar 

  7. Duffie D, Kan R (1996) A yield-factor model of interest rates. Math Finance 6:379–406

    Article  MATH  Google Scholar 

  8. Elliott R, Aggoun L, Moore J (1995) Hidden Markov models: estimation and control. Springer, New York

    MATH  Google Scholar 

  9. Elliott R, Mamon R (2002) An interest rate model with a Markovian mean-reverting level. Quant Finance 2:454–458

    Article  MathSciNet  Google Scholar 

  10. Elliott R, Wilson C (2007) The term structure of interest rates in a hidden Markov setting. In: Mamon R, Elliott R (eds) Hidden Markov Models in Finance, Springer, pp 15–30

  11. Elliott R, Siu T (2009) On Markov-modulated exponential-affine bond price formulae. Appl Math Finance 16:1–15

    Article  MathSciNet  MATH  Google Scholar 

  12. Fan K, Shen Y, Siu T, Wang R (2015) Pricing annuity guarantees under a double regime-switching model. Insur: Math Econ (in press)

  13. Gao H, Mamon R, Liu X, Tenyakov A (2015) Mortality modelling with regime-switching for the valuation of a guaranteed annuity option. Insur: Math Econ 63:108–120

    MathSciNet  Google Scholar 

  14. Jalen L, Mamon R (2009) Valuation of contingent claims with mortality and interest rate risks. Math Comput Model 49:1893–1904

    Article  MathSciNet  MATH  Google Scholar 

  15. Lee R, Carter L (1992) Modeling and forecasting US mortality. J Am Stat Assoc 87:659–675

    Google Scholar 

  16. Liu X, Mamon R, Gao H (2013) A comonotonicity-based valuation method for guaranteed annuity options. J Comput Appl Math 250:58–69

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu X, Mamon R, Gao H (2014) A generalised pricing framework addressing correlated mortality and interest risks: a change of probability measure approach. Stochastics 86:594–608

    MathSciNet  Google Scholar 

  18. Longstaff F, Schwartz E (1992) Interest rate volatility and the term structure: a two-factor general equilibrium model. J Finance 47(4):1259–1282

    Article  Google Scholar 

  19. Luciano E, Vigna E (2008) Non mean reverting affine processes for stochastic mortality. Belg Actuar Bull 8:5–16

    MathSciNet  Google Scholar 

  20. Mamon R (2002) A time-varying Markov chain model of term structure. Stat Probab Lett 60(3):309–312

    Article  MathSciNet  MATH  Google Scholar 

  21. Mamon R (2004) Three ways to solve for bond prices in the Vasiček model. J Appl Math Decis Sci 8:1–14

    Article  MathSciNet  MATH  Google Scholar 

  22. Mamon R, Elliott R (2014) Hidden Markov models in finance: further developments and applications. Springer, New York

    Book  Google Scholar 

  23. Rodrigo M, Mamon R (2014) An alternative approach to the calibration of the Vasiček and CIR interest rate models via generating functions. Quant Finance 14:1961–1970

    Article  MathSciNet  Google Scholar 

  24. Shen Y, Siu T (2013) Longevity bond pricing under stochastic interest rate and mortality with regime-switching. Insur: Math Econ 52:114–123

    MathSciNet  MATH  Google Scholar 

  25. Wilkie A, Waters H, Yang S (2003) Reserving, pricing and hedging for policies with guaranteed annuity options. Br Actuar J 9:263–425

    Article  Google Scholar 

  26. Xi X, Mamon R (2011) Parameter estimation of an asset price model driven by a weak hidden Markov chain. Econ Model 28(2011):36–46

    Article  Google Scholar 

  27. Xi X, Mamon R, Davison M (2014) A higher-order hidden Markov chain-modulated model for asset allocation. J Math Model Algorithms Oper Res 13(1):59–85

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors would like to express their appreciation to the anonymous referee and co-editor for their encouragement and helpful comments.

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Authors and Affiliations

  1. Department of Statistical and Actuarial Sciences, University of Western Ontario, 1151 Richmond Street, London, ON, N6A 5B7, Canada

    Huan Gao, Rogemar Mamon & Xiaoming Liu

Authors
  1. Huan Gao
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  2. Rogemar Mamon
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  3. Xiaoming Liu
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Corresponding author

Correspondence to Rogemar Mamon.

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Gao, H., Mamon, R. & Liu, X. Pricing a guaranteed annuity option under correlated and regime-switching risk factors. Eur. Actuar. J. 5, 309–326 (2015). https://doi.org/10.1007/s13385-015-0118-3

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  • DOI: https://doi.org/10.1007/s13385-015-0118-3

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