Valuation of Equity-Linked Life Insurance Contracts with Flexible Guarantees in a Non Gaussian Economy

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 risky assets. The first one has larger expected returns but is riskier while the second one is less risky but still can earn more than an investment in a risk-free asset. Of course this payoff can be seen as the result of an investment in the first asset and a long position in an exchange option. The latter was priced under Gaussian assumptions by Margrabe (1978). To take kurtosis into account the underlying dynamics have to be changed. In this paper, we suggest modeling the underlying dynamics of the second asset by a simple diffusion, i.e. a geometric Brownian motion 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 manage 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 also consider a contract with a fixed guarantee and price it while taking into account stochastic volatility.