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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:

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