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**Equity-linked life insurance: A model with stochastic interest rates.**
*(English)*
Zbl 0872.62094

Summary: Assuming constant interest rate M. J. Brennan and E. S. Schwartz [J. Fin. Econ. 3, 195-213 (1976)] obtained the rational insurance premium on an equity-linked insurance contract through the application of the theory of contingent claims pricing. Further, considerations with deterministic interest rate have been discussed by K. K. Aase and S. A. Persson. Analysing the single premium case A. R. Bacinello and F. Ortu [Trans. 3rd AFIR Int. Coll., 35-55 (1993)] allow for the short term interest rate to develop in accordance to an Ornstein-Uhlenbeck process. In a later paper [L. Peccati and M. Viren (eds.), Financial Modelling, 1-25 (1994)] they consider extensions to both the single and the periodic premium model.

This paper presents a model similar to the one by Bacinello and Ortu (1994) for the periodic premium case with stochastic interest rate dynamics. It is shown that the insurance contract includes an Asian-like option contract. Sufficient conditions on the guaranteed amount for the existence of a solution are derived. As no closed form solution will be obtained, we discuss different numerical approaches and apply Monte Carlo simulations with a variance reduction technique.

This paper presents a model similar to the one by Bacinello and Ortu (1994) for the periodic premium case with stochastic interest rate dynamics. It is shown that the insurance contract includes an Asian-like option contract. Sufficient conditions on the guaranteed amount for the existence of a solution are derived. As no closed form solution will be obtained, we discuss different numerical approaches and apply Monte Carlo simulations with a variance reduction technique.

### MSC:

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

### Keywords:

forward risk adjusted measure; periodic premium case; stochastic interest rate dynamics; Asian-like option contract; Monte Carlo simulations; variance reduction
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\textit{J. A. Nielsen} and \textit{K. Sandmann}, Insur. Math. Econ. 16, No. 3, 225--253 (1995; Zbl 0872.62094)

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### References:

[1] | Aase, K. K.; Persson, S. A., (Pricing of Unit-Linked Life Insurance Policies (1992), WP, Norwegian School of Economics and Business Administration: WP, Norwegian School of Economics and Business Administration Bergen) |

[2] | Bacinello, A. R.; Ortu, F., Pricing equity-linked life insurance with endogenous minimum guarantees, Insurance: Mathematics and Economics, 12, 245-257 (1993) · Zbl 0778.62093 |

[3] | Bacinello, A. R.; Ortu, F., Pricing guaranteed securities-linked life insurance under interest-rate risk, (Actuarial Approach For Financial Risks. Actuarial Approach For Financial Risks, Transactions of the 3rd AFIR International Colloquium (1993)), 35-55 |

[4] | Bacinello, A. R.; Ortu, F., Single and periodic premiums for guaranteed equity-linked life insurance under interest-rate risk: the lognormal+Vasicek case, (Peccati, L.; Viren, M., Financial Modelling (1994), Physica-Verlag), 1-25 |

[5] | Brennan, M. J.; Schwartz, E. S., The pricing of equity-linked life insurance policies with an asset value guarantee, Journal of Financial Economics, 3, 195-213 (1976) |

[6] | Brennan, M. J.; Schwartz, E. S., Pricing and investment strategies for equity-linked life insurance policies with an asset value guarantee, Journal of Financial Economics, 3, 195-213 (1979) |

[7] | Brennan, M. J.; Schwartz, E. S., Pricing and investment strategies for equity-linked life insurance, (Huebner Foundation Monograph, 7 (1979), Wharton School, University of Pennsylvania: Wharton School, University of Pennsylvania Pennsylvania) |

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[9] | Delbean, F., Equity linked policies, Bulletin Association Royal Actuaires Belge, 33-52 (1990) |

[10] | Ho, T. S.Y.; Lee, S., Term structure movements and pricing interest rate contingent claims, Journal of Finance, 41, 1011-1029 (1986) |

[11] | Kemma, A. G.Z.; Vorst, A. C.F., A pricing method for options based on average asset values, Journal of Banking and Finance, 14, 113-129 (1990) |

[12] | Persson, S.-A., Valuation of a multistate life insurance contract with random benefit, The Scandinavian Journal of Management, 9S, 73-86 (1993) |

[13] | Reimer, M., Arithmetic-average-price options: Bewertungsverfahren und Simulationsstudie, (Discussion Paper B-240 (1993), University of Bonn) |

[14] | Turnbull, S. M.; Wakeman, L. M., Quick algorithm for pricing European average options, Journal of Financial and Quantitative Analysis, 77-389 (1991) |

[15] | Vorst, A. C.F., Prices and hedge ratios of average exchange rate options, International Review of Financial Analysis, 1, 179-193 (1992) |

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