Efficient valuation of equity-indexed annuities under Lévy processes using Fourier cosine series

Equity-indexed annuities are deferred annuities that accumulate value over time according to crediting formulas and realized equity index returns. We propose an efficient algorithm to value two popular crediting formulas found in equity-indexed annuities – annual point-to-point (APP) and monthly point-to-point (MPP) – under general Lévy-process-based index returns. APP contracts observe returns of referenced indexes annually and credit equity-indexed annuity accounts, subject to minimum and maximum returns. MPP contracts incorporate both local/monthly caps and global/annual floors on index credits. MPP contracts have payoffs of a “cliquet†option. Our algorithm, based on the COS method of Fang and Oosterlee, expands the present value of an equity-indexed annuity contract using Fourier cosine series, and expresses its value as a series of terms involving simple characteristic function evaluations. We present several examples with different Lévy processes, including the Black–Scholes model and the Carr–Geman–Madan–Yor model. These examples illustrate the efficiency of our algorithm, as well as its versatility in computing annuity market sensitivities, which could facilitate the hedging and pricing of annuity contracts.