Impact of Stochastic Interest Rates and Stochastic Volatility on Variable Annuities

With the success of variable annuities, insurance companies are piling up large risks in terms of both equity and fixed income assets. These risks should be properly modeled as the resulting dynamic hedging strategy is very sensitive to the modeling assumptions. The current literature has been largely focusing on simple variations around Black-Scholes model with basic interest rates term structure models. However, in a more realistic world, one should account for both Stochastic Volatility and Stochastic Interest rates. In this paper, we examine the combine effect of a Heston-type model for the underlying asset with a HJM affine stochastic interest rates model and apply it to the pricing of GMxB (GMIB, GMDB, GMAB and GMWB). We see that stochastic volatility and stochastic interest rates have an impact on the resulting fair value of the contract and the resulting fair fee as well as mainly on the vega hedge. Interestingly, using a stochastic volatility model leads to scenarios with high level of volatility for long maturities resulting in a higher contract value and a resulting fair fee. We also see that the impact of stochastic interet rates and volatility is more pronounced on the vega hedge than on the delta hedge.