## 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)

### Keywords:

equity linked death benefits; PROJ; GMDB
Full Text:

### References:

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