Valuing guaranteed withdrawal benefits with stochastic interest rates and volatility

Guaranteed withdrawal benefits are long term contracts which provide investors with equity participation while guaranteeing them a secured income stream. Due to the long investment horizons involved, stochastic volatility and stochastic interest rates are important factors to include in their valuation. Moreover, investors are typically allowed to participate in a mixed fund composed of both equity and fixed-income securities. Here, we develop an efficient method for valuing these path-dependent products through re-writing the problem in the form of an Asian styled claim and a dimensionally reduced partial differential equation (PDE). The PDE is then solved using an Alternating Direction Implicit method. Furthermore, we derive an analytical closed form approximation and compare this approximation with the PDE results and find excellent agreement. We illustrate the various effects of the parameters on the valuation through numerical experiments and discuss their financial implications.