## Convex upper and lower bounds for present value functions.(English)Zbl 0971.91030

The authors consider the present value of a series of $$n$$ payments $$c_{i}$$ at times $$\tau_{i}, i=1,\ldots,n$$ in the form $$V_0= \sum_{i=1}^{n}c_{i}\exp(-\int_{0}^{\tau_{i}} r(s) ds)$$. Here the instantaneous interest rate $$r(t)$$ satisfies either the Vasicek model stochastic differential equation $$dr=(\alpha-\beta r) dt+ \gamma dB$$, where $$\alpha,\beta,\gamma\geq 0$$, $$B(t)$$ is the standard Wiener process, or the Ho-Lee model $$dr=\alpha(t) dt+\gamma dB$$. In this paper analytical upper and lower bounds (in convex order sense) for the distribution function of $$V_0$$ are obtained. The accuracy of the proposed bounds is investigated by comparing their cumulative distribution functions to the empirical cumulative distribution functions obtained by Monte Carlo simulation.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 91B28 Finance etc. (MSC2000)
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### References:

 [1] Vasicek, Journal of Financial Economics 5 pp 177– (1977) · Zbl 1372.91113 [2] Ho, Journal of Finance 41 pp 1011– (1986) [3] Effective Actuarial Methods. North-Holland: Amsterdam, Insurance Series 3, 1990. [4] Stochastic Orders and Their Applications, Academic Press: New York, 1994. · Zbl 0806.62009 [5] Comonotonicity and maximal stop-loss premiums. Bulletin of the Swiss Association of Actuaries 2000. To appear. [6] Goovaerts, Journal of Risk and Insurance 67 pp 1– (2000) [7] Goovaerts, Insurance: Mathematics and Economics 24 pp 281– (1999) [8] Non-Additive Measure and Integral. Kluwer Academic Publishers: Boston, 1994. · Zbl 0826.28002 [9] Simon, Insurance: Mathematics & Economics 26 pp 175– (2000) [10] Rogers, Journal of Applied Probability 32 pp 1077– (1995) [11] Quantum Mechanics and Path-Integrals. Mc-Graw Hill: New York, 1965. [12] The discretization bias for processes of the short-term interest rate: An empirical analysis, http://staff.fucam.ac.be/?dewinne/ [12 October 2000].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.