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Pricing Dynamic Investment Fund Protection

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  • Hans Gerber
  • Gérard Pafumi

Abstract

We consider an investment fund whose unit value is modeled by a geometric Brownian motion. Different forms of dynamic investment fund protection are examined. The basic form is a guarantee that instantaneously provides the necessary payments so that the upgraded fund unit value does not fall below a protected level. A closed form expression for the price of such a guarantee is derived. This result can also be applied to price a guarantee where the protected level is an exponential function of time. Moreover, it is explicitly shown how the protection can be generated by construction of the replicating portfolio. The dynamic investment fund protection is compared with the corresponding put option, and it is shown that for short time intervals the ratio of the prices approaches 2. Finally, a more exotic guarantee is considered, where the protected level is a given percentage of the maximal observed fund unit value. Assuming that the protected level remains constant once the payments have started, we obtain a surprisingly simple formula for the price of a perpetual guarantee. Several numerical and graphical illustrations show how the theoretical results can be implemented in practice.

Suggested Citation

  • Hans Gerber & Gérard Pafumi, 2000. "Pricing Dynamic Investment Fund Protection," North American Actuarial Journal, Taylor & Francis Journals, vol. 4(2), pages 28-37.
    • Handle: RePEc:taf:uaajxx:v:4:y:2000:i:2:p:28-37
    DOI: 10.1080/10920277.2000.10595894
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    Cited by:

    1. Dean Buckner & Kevin Dowd & Hardy Hulley, 2022. "Arbitrage Problems with Reflected Geometric Brownian Motion," Papers 2201.05312, arXiv.org, revised Sep 2022.
    2. repec:dau:papers:123456789/5374 is not listed on IDEAS
    3. Markus Hertrich, 2022. "Foreign exchange interventions under a minimum exchange rate regime and the Swiss franc," Review of International Economics, Wiley Blackwell, vol. 30(2), pages 450-489, May.
    4. El Karoui, Nicole & Jeanblanc, Monique & Lacoste, Vincent, 2005. "Optimal portfolio management with American capital guarantee," Journal of Economic Dynamics and Control, Elsevier, vol. 29(3), pages 449-468, March.
    5. Moore, Kristen S., 2009. "Optimal surrender strategies for equity-indexed annuity investors," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 1-18, February.
    6. Markus Hertrich & Heinz Zimmermann, 2017. "On the Credibility of the Euro/Swiss Franc Floor: A Financial Market Perspective," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 49(2-3), pages 567-578, March.
    7. Hertrich Markus, 2016. "The Costs of Implementing a Unilateral One-Sided Exchange Rate Target Zone," Review of Economics, De Gruyter, vol. 67(1), pages 91-120, May.
    8. Linyi Qian & Zhuo Jin & Wei Wang & Lyu Chen, 2018. "Pricing dynamic fund protections for a hyperexponential jump diffusion process," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(1), pages 210-221, January.
    9. Chu, Chi Chiu & Kwok, Yue Kuen, 2004. "Reset and withdrawal rights in dynamic fund protection," Insurance: Mathematics and Economics, Elsevier, vol. 34(2), pages 273-295, April.
    10. Jun Sekine, 2012. "Long-term optimal portfolios with floor," Finance and Stochastics, Springer, vol. 16(3), pages 369-401, July.
    11. Han, Heejae & Jeon, Junkee & Kang, Myungjoo, 2016. "Pricing chained dynamic fund protection," The North American Journal of Economics and Finance, Elsevier, vol. 37(C), pages 267-278.
    12. 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.
    13. R. Guy Thomas, 2023. "Long-term option pricing with a lower reflecting barrier," Papers 2302.05808, arXiv.org.
    14. Gerber, Hans U. & Shiu, Elias S.W. & Yang, Hailiang, 2012. "Valuing equity-linked death benefits and other contingent options: A discounted density approach," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 73-92.

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