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Pricing Lookback Options and Dynamic Guarantees

Author

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  • Hans Gerber
  • Elias Shiu

Abstract

Pricing exotic options or guarantees in equity-indexed annuities can be problematic. The authors present closed-form formulas for pricing lookback options and dynamic guarantees that facilitate the hedging and reserving for such products. The principal tool used is a closed-form expression for B(u, T), the Laplace-Stieltjes transform of the expected excess of the running maximum of a Wiener process above a positive constant u in a finite time interval of length T. If the aggregate net income of a company is modeled with a Wiener process, then the excess of the running maximum above u can be interpreted as aggregate dividend payments, and the quantity B(u, T) is the expectation of the discounted value of the dividend payments up to time T. The formula for B(u, T) is used to price European lookback options (call and put, fixed and floating strike). It is also used to price dynamic fund protection, which is a guarantee on an investment fund: The number of units of the investment fund is increased whenever necessary, so that their total value does not fall below a guaranteed level. The guaranteed level can be stochastic, such as that given by a stock index. Some well-known results for the first passage time of the Wiener process are explained in the appendix.

Suggested Citation

  • Hans Gerber & Elias Shiu, 2003. "Pricing Lookback Options and Dynamic Guarantees," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(1), pages 48-66.
    • Handle: RePEc:taf:uaajxx:v:7:y:2003:i:1:p:48-66
    DOI: 10.1080/10920277.2003.10596076
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    Citations

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    Cited by:

    1. Chiu, Yu-Fen & Hsieh, Ming-Hua & Tsai, Chenghsien, 2019. "Valuation and analysis on complex equity indexed annuities," Pacific-Basin Finance Journal, Elsevier, vol. 57(C).
    2. Lee, Hangsuck & Ha, Hongjun & Lee, Minha, 2023. "Partial quanto lookback options," The North American Journal of Economics and Finance, Elsevier, vol. 64(C).
    3. Gao, Yin & Jia, Lifen, 2021. "Pricing formulas of barrier-lookback option in uncertain financial markets," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    4. Lee, Hangsuck & Ha, Hongjun & Kim, Eunchae & Lee, Minha, 2024. "Quanto fund protection using partial lookback participation," The North American Journal of Economics and Finance, Elsevier, vol. 73(C).
    5. Alexander Bohnert, 2015. "The Impact of Guarantees on the Performance of Pension Saving Schemes: Insights from the Literature," Risks, MDPI, vol. 3(4), pages 1-28, November.
    6. L. Ramprasath, 2018. "A simpler algorithm to price American Lookback options in a discrete stochastic volatility model," Working papers 294, Indian Institute of Management Kozhikode.
    7. Gan Guojun & Valdez Emiliano A., 2017. "Valuation of large variable annuity portfolios: Monte Carlo simulation and synthetic datasets," Dependence Modeling, De Gruyter, vol. 5(1), pages 354-374, December.
    8. Kijima, Masaaki & Wong, Tony, 2007. "Pricing of Ratchet equity-indexed annuities under stochastic interest rates," Insurance: Mathematics and Economics, Elsevier, vol. 41(3), pages 317-338, November.
    9. Sarah Dendievel & Guy Latouche, 2017. "Approximations for Time-Dependent Distributions in Markovian Fluid Models," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 285-309, March.
    10. Qian, Linyi & Wang, Wei & Wang, Rongming & Tang, Yincai, 2010. "Valuation of equity-indexed annuity under stochastic mortality and interest rate," Insurance: Mathematics and Economics, Elsevier, vol. 47(2), pages 123-129, October.
    11. Kokou Essiomle & Franck Adékambi, 2023. "Valuation of Equity-Linked Death Benefits on Two Lives with Dependence," Risks, MDPI, vol. 11(1), pages 1-26, January.
    12. 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.
    13. Gan, Guojun & Lin, X. Sheldon, 2015. "Valuation of large variable annuity portfolios under nested simulation: A functional data approach," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 138-150.
    14. Lee, Hangsuck & Ha, Hongjun & Lee, Minha, 2022. "Foreign equity lookback options with guarantees," Finance Research Letters, Elsevier, vol. 48(C).
    15. Gan, Guojun, 2013. "Application of data clustering and machine learning in variable annuity valuation," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 795-801.
    16. 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.
    17. Orozco-Garcia, Carolina & Schmeiser, Hato, 2015. "How sensitive is the pricing of lookback and interest rate guarantees when changing the modelling assumptions?," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 77-93.
    18. Lee, Hangsuck & Kim, Eunchae & Ko, Bangwon, 2022. "Valuing lookback options with barrier," The North American Journal of Economics and Finance, Elsevier, vol. 60(C).
    19. V. M. Belyaev, 2011. "Pricing Variable Annuity Contracts with High-Water Mark Feature," Papers 1108.4393, arXiv.org, revised Aug 2011.

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