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.


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