Analytical approximations for the value of embedded options in unit-linked insurance and profit sharing

Abstract

Many insurance companies sell products that involve embedded options. The value of such an option represents the expected future liability and therefore it is important that insurers can value the options they sold. Since most of these options are very complex, they are valued using Monte Carlo simulations. This requires considerable computation resources and therefore methods have been developed to approx- imate the option values analytically. In this thesis we study two of these options and give analytical approximations for their values. The first option is a guarantee in unit-linked insurance for which upper and lower bounds are derived using the concept of comonotonicity as developed by Dhaene, Denuit, Goovaerts, Kaas and Vyncke (2002a, 2002b). This is done in the Black-Scholes model as well as in the Hull-White-Black-Scholes model, where the latter has the additional feature of stochastic interest rates. The lower bound is the same as derived in Schrager and Pelsser (2004), but the derivation by explicitly applying the concept of comonotonicity was not given before. The second option is a call option on an average of swap rates as used in profit sharing. The value is approximated by using approximate swap rate dynamics as developed by Schrager and Pelsser (2006). Finally, the quality of the approximations is determined by comparing them to Monte Carlo simulations. It turns out that the lower bound for the guarantee in unit-linked insurance is a very accurate approximation.