Valuation of Bivariate Minimum Guarantees through Option Modelling and Copulas

Pension plans and life insurances oering minimum performance guarantees are very com- mon worldwide. In some markets, for example the brazilian case, besides the minimum guar- anteed rate, the costumers of some dened contribution plans have the right to receive, over their savings, the positive dierence between the return of a specied investment fund, usually a xed income fund, and the minimum guaranteed rate, commonly dened as the composition of a xed interest rate and a oating ination rate. This instrument can be characterized as an option to exchange one asset, the minimum guaranteed rate, for another, the return of the specied investment fund. In this paper, we provide a closed formula to evaluate this liabil- ity that depends on two stochastic rates assuming bivariate normality. We also provide some examples assuming others copulas functions, using Monte Carlo simulation, and compare the eects of the copula and marginals specication in the price of the option. The model makes use of a one-factor Vasicek framework for the term structures of interest rate and ination rate.