Analytical Pricing of An Insurance Embedded Option: Alternative Formulas and Gaussian Approximation

Abstract

Analytical pricing of a double-trigger option with the Black-Scholes-Vasicek (BSV) state price deflator is considered. In the context of market-consistent valuation of insurance liabilities, the option appears as embedded option of index-linked endowment policies that provide combined protection against inflation and a minimum interest rate guarantee by death. A first analytical pricing formula in terms of the standard bivariate normal distribution is derived. Then, using alternatively the canonical BSV deflator, a second integral representation is derived. Based on an elementary Gaussian integral in three variables a third integral decomposition is obtained and approximated by closed-form Gaussian expressions using a simple approximation by Lin of the normal tail probability integral. Similarly to the invariance of the Black-Scholes and Margrabe formulas with respect to the market prices of the risk factors, two of the alternative double-trigger option pricing formulas and the proposed Gaussian approximation also share this property. A numerical example rounds up the analysis by showing accuracy of the Gaussian approximation within some few negligible basis points.