A flexible lattice framework for valuing options on assets paying discrete dividends and variable annuities embedding GMWB riders

We propose a lattice framework that allows to obtain accurate evaluation and management of the risk affecting both long-term options written on assets paying discrete dividends, and variable annuities (VAs hereafter) embedding some provisions, like a guaranteed minimum withdrawal benefit (GMWB hereafter). To consider a realistic context that permits to achieve consistent risk estimates, the model has been developed in a market where the traded asset dynamics is characterized by stochastic interest. In addition, the model is flexible in that it allows to combine financial and demographic risk, to embed in the contract early exercise features, and to choose the most appropriate dynamics both for the interest rates and the traded assets among the ones widely used in finance and insurance. The proposed framework is general because it may accommodate a wide range of diffusions widely used in finance and insurance, both for the stochastic interest rate and for the underlying activity, by simply modifying the process parameters. The considered continuous time processes are approximated by means of recombining lattices, which work directly on the original process. Indeed, the generated lattice values discretize only the diffusion part of the processes, while branching probabilities are computed to assure that the first and the second order local moments of the discrete distribution match the corresponding continuous-time ones, at least within the limit. After discretizing the spot rate process, the algorithm works to discretize the dividend-paying underlying asset process in the case we want to evaluate an option, or the sub-account value in the case we want to evaluate a VAs with GMWBs. The discretization is still based on a recombining binomial lattice that is similar to the one used to approximate the interest rate dynamics but, in addition, presents a vector of representative values in correspondence with each node, generated by taking into account the payments of the dividends or withdrawals up to that node. The resulting two lattices are combined in order to establish a bivariate tree presenting four branches for each node. Joint probabilities are computed for the possible jumps in order to take into account the correlation between the two processes. A linear interpolation technique is used when solving backward through the lattice to compute the option price or the VA present value in terms of the discounted policy payoffs over the bivariate lattice branches.