Variable annuity with a surrender option under multiscale stochastic volatility

In this paper, we study the valuation of a variable annuity embedding an early surrender option in which the underlying equity price follows the multiscale stochastic volatility model. Utilizing singular and regular perturbation techniques by Fouque et al. (in Multiscale Stochastic volatility for equity, interest rate, and credit derivatives. Cambridge University Press: Cambridge, 2011), we provide a closed-form solution of the valuation in an asymptotical sense. Comparison with another simulation-based method is presented to confirm the accuracy of our valuation methodology. We show that the fair insurance fee of the variable annuity with the surrender option far more decreases in an increase of the underlying equity price than the fair insurance fee of the annuity without the option does. The multiscale model's conventional feature (i.e., the fast and slow time scale volatilities are influential in the short-term and long-term products, respectively) is observed in the fair insurance fee of the annuity without the surrender option. When the annuity contract embeds the surrender option, however, the effects of the fast scale volatility on the fair insurance fee becomes more remarkable.