Surrender and path-dependent guarantees in variable annuities: integral equation solutions and benchmark methods

We investigate the evaluation problem of variable annuities by considering guaranteed minimum maturity benefits with constant or path-dependent guarantees of barrier and lookback type. We propose to solve the non-standard Volterra integral equations associated with the policy valuations through a randomized trapezoidal quadrature rule combined with an interpolation technique. Such a rule improves the convergence rate with respect to the classical trapezoidal quadrature, while the interpolation technique allows us to obtain an efficient algorithm that produces an accurate approximation of the early exercise boundary. The method accuracy is assessed by constructing two benchmarks based on lattice approaches and the least-squares Monte Carlo simulations. In the first case, a novel algorithm for the lookback path-dependent guarantee is obtained thanks to the lattice convergence properties.