Abstract
In this paper, we focus on the pricing of a particular life insurance contract where the conditional payoff to the policyholder is the maximum of two assets. We begin with the case of a Guaranteed Minimum Maturity Benefit contract where the guarantee is certain, and in a second part we consider a flexible guarantee leading to what we call a Pure Endowment Flexible Guarantee contract. In this latter case, both assets are risky. The first one has larger expected returns but is riskier while the second one is less risky but can still earn more than an investment in the risk-free asset. To take kurtosis into account the underlying dynamics have to be changed. In this paper, we suggest modelling the underlying dynamics of the second asset with a simple diffusion, i.e. a geometric Brownian otion with a low volatility, while the riskier asset follows a jump diffusion. More precisely, this process has a Brownian component and a compound Poisson one, where jump size is driven by a double exponential distribution. This stochastic process introduced by Kou (2002) is easy to use and proves to be a versatile tool. To price our life insurance contract, we use a generalized Fourier transform and obtain the solution numerically. As far as we know, this is the first paper to use this approach. This methodology proves to be very efficient both with respect to accuracy and to computational time. We incorporate mortality using a classical Makeham law.