Insurance: Mathematics and Economics 45 (2009) 286–295 Contents lists available at ScienceDirect Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime Evaluating fair premiums of equity-linked policies with surrender option in a bivariate model ✩ Massimo Costabile a,∗ , Marcellino Gaudenzi b , Ivar Massabò a , Antonino Zanette b a Dipartimento di Scienze Aziendali, Università della Calabria, Ponte Bucci, Cubo 3C, 87036 Rende (CS), Italy b Dipartimento di Finanza dell’Impresa e dei Mercati Finanziari, Università di Udine, Via Tomadini 30/A, 33100 Udine, Italy article info Article history: Received April 2009 Received in revised form July 2009 Accepted 27 July 2009 Insurance branch category: IB11 JEL classification: G22 Subject category: IM01 IM30 Keywords: Equity-linked policies Bivariate model Surrender option abstract We tackle the problem of computing fair periodical premiums of an equity-linked policy with a maturity guarantee and an embedded surrender option. We consider the policy as a Bermudan-style contingent claim that can be exercised at the premium payment dates. The evaluation framework is based on a discretization of a bivariate model that considers the joint evolution of the equity value with stochastic interest rates. To deeply reduce the computational complexity of the pricing problem we use the singular points framework that allows us to compute accurate upper and lower estimates of the policy premiums. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Nowadays equity-linked policies have gained a wide popularity in insurance market. The main reason is that they give the opportunity to link the capital invested into the policy to the performance of a portfolio of equities. In this way, the coverage provided in case of death or survivance of the insured is coupled with the possibility to obtain higher returns from the capital market than those guaranteed by traditional policies. The problem is that in the latter case the insured bears the risk of a negative performance of the equities considered. To mitigate this risk, insurance companies usually insert into the contract a minimum guarantee that assures the policyholder to receive at maturity (or before in the case of early termination of the contract) at least a prespecified sum. ✩ This research was supported by MIUR (Prin 2007). ∗ Corresponding author. Tel.: +39 0984492258; fax: +39 0984492277. E-mail addresses: massimo.costabile@unical.it (M. Costabile), gaudenzi@uniud.it (M. Gaudenzi), i.massabo@unical.it (I. Massabò), antonino.zanette@uniud.it (A. Zanette). In their seminal papers, Brennan and Schwartz (1976) and Boyle and Schwartz (1977) applied financial mathematics techniques, developed in a Black–Scholes framework to compute fair pre- miums of equity-linked policies with minimum guarantee. Since then, a huge number of contributions tackled the fair pre- mium evaluation problem in several directions. Among others we mention Aase and Persson (1994) and Delbaen (1990) that con- sidered periodical premiums computed via Monte Carlo simula- tions. Bacinello and Ortu (1993), at first, extend these models to the case in which the minimum guarantee is endogenous, i.e., it is a function of the premium(s) paid. Then, they derived a closed form formula for the single premium of an endowment equity-linked policy in a framework where interest rates are stochastic (Bacinello and Ortu, 1994). Nielsen and Sandmann (1995) developed a model with stochastic interest rates to evaluate periodic premiums com- puted using numerical procedures. Often, an equity-linked policy embeds a surrender option that gives the policyholder the chance to escape out of the contract before maturity. The fair premiums evaluation problem may consider the decision to surrender the contract as exogenous. In this case, together with death, early withdrawal is considered as a second exogenous cause of early termination of the contract. Then, 0167-6687/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2009.07.008