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Valuing equity-linked death benefits in general exponential Lévy models. (English) Zbl 1430.91079
Let \(\tau\) be the remaining life time of an insured. \(\tau\) is assumed to be an absolutely continuous variable. The life insurance policy is linked to an index \(S(t) = S(0) e^{X(t)}\), where \(\{X(t)\}\) is a Lévy process. The benefit at the time of death is \(b(S(\tau))\). One is therefore interested in \(V(T) = \mathbb{E}[e^{-\delta \tau} b(S(\tau)) I_{\tau < T}]\). Here \(T \in (0,\infty]\). A finite \(T\) corresponds to a contract with an expiring date. It is now assumed that \(b(s) = s^c I_{D}(s)\) for some \(c \ge 0\) and \(D \subset \mathbb{R}\). For example, the put and call options are differences between two such functions with \(c \in \{0,1\}\). It is further assumed that \(|V(\infty)| < \infty\).
The value of the death benefit can then be expressed as \[ S^c(0) \mathbb{E}[e^{-\delta \tau} e^{c X(\tau)} I_{D}(S(0) e^{X(\tau)})]\;.\] The function under the expectation is now projected on the set generated by the functions \(\varphi_{a,n}(\cdot) = a^{\frac12} \varphi(a(\cdot-\xi_n))\), where \(\varphi\) is a generator function with finite support, \(\xi_n = \xi_0 + n/a\), and \(a > 0\) is a bandwidth. Two examples for the generator \(\varphi\) are used; \(\tilde \varphi^{*k}(x)\) (\(k\)-th convolution) for \(k=2,3\) and \(\tilde \varphi(x)\) is the density of the uniform(0,1) distribution. The expression looked for has then the form \(S^c(0) \sum_{n=0}^\infty A_{n,c} \int_G \varphi_{a,n}(x) \;d x\), where \(G\) is the set corresponding to \(D\). For the approximation, the finite sum and a finite set \(G\) is chosen. The error of the approximation is analysed and the theory is illustrated by some numerical examples.

MSC:
91G05 Actuarial mathematics
60G51 Processes with independent increments; Lévy processes
91G60 Numerical methods (including Monte Carlo methods)
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