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Valuing equity-linked death benefits and other contingent options: A discounted density approach

  • Gerber, Hans U.
  • Shiu, Elias S.W.
  • Yang, Hailiang
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    Motivated by the Guaranteed Minimum Death Benefits in various deferred annuities, we investigate the calculation of the expected discounted value of a payment at the time of death. The payment depends on the price of a stock at that time and possibly also on the history of the stock price. If the payment turns out to be the payoff of an option, we call the contract for the payment a (life) contingent option. Because each time-until-death distribution can be approximated by a combination of exponential distributions, the analysis is made for the case where the time until death is exponentially distributed, i.e., under the assumption of a constant force of mortality. The time-until-death random variable is assumed to be independent of the stock price process which is a geometric Brownian motion. Our key tool is a discounted joint density function. A substantial series of closed-form formulas is obtained, for the contingent call and put options, for lookback options, for barrier options, for dynamic fund protection, and for dynamic withdrawal benefits. In a section on several stocks, the method of Esscher transforms proves to be useful for finding among others an explicit result for valuing contingent Margrabe options or exchange options. For the case where the contracts have a finite expiry date, closed-form formulas are found for the contingent call and put options. From these, results for De Moivre’s law are obtained as limits. We also discuss equity-linked death benefit reserves and investment strategies for maintaining such reserves. The elasticity of the reserve with respect to the stock price plays an important role. Whereas in the most important applications the stopping time is the time of death, it could be different in other applications, for example, the time of the next catastrophe.

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    Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

    Volume (Year): 51 (2012)
    Issue (Month): 1 ()
    Pages: 73-92

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    Handle: RePEc:eee:insuma:v:51:y:2012:i:1:p:73-92
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    1. Bacinello, Anna Rita & Millossovich, Pietro & Olivieri, Annamaria & Pitacco, Ermanno, 2011. "Variable annuities: A unifying valuation approach," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 285-297.
    2. Lee, Hangsuck, 2003. "Pricing equity-indexed annuities with path-dependent options," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 677-690, December.
    3. Gerber, Hans U. & Shiu, Elias S. W., 1996. "Actuarial bridges to dynamic hedging and option pricing," Insurance: Mathematics and Economics, Elsevier, vol. 18(3), pages 183-218, November.
    4. Goldman, M Barry & Sosin, Howard B & Gatto, Mary Ann, 1979. "Path Dependent Options: "Buy at the Low, Sell at the High"," Journal of Finance, American Finance Association, vol. 34(5), pages 1111-27, December.
    5. Shang, Zhaoning & Goovaerts, Marc & Dhaene, Jan, 2011. "A recursive approach to mortality-linked derivative pricing," Insurance: Mathematics and Economics, Elsevier, vol. 49(2), pages 240-248, September.
    6. Eric R. Ulm, 2006. "The Effect of the Real Option to Transfer on the Value of Guaranteed Minimum Death Benefits," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 73(1), pages 43-69.
    7. Ko, Bangwon & Shiu, Elias S.W. & Wei, Li, 2010. "Pricing maturity guarantee with dynamic withdrawal benefit," Insurance: Mathematics and Economics, Elsevier, vol. 47(2), pages 216-223, October.
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