Valuation of guaranteed annuity options using a stochastic volatility model for equity prices

Guaranteed Annuity Options are options providing the right to convert a policyholder’s accumulated
funds to a life annuity at a fixed rate when the policy matures. These options were a
common feature in UK retirement savings contracts issued in the 1970’s and 1980’s when interest
rates were high, but caused problems for insurers as the interest rates began to fall in the 1990’s.
Currently, these options are frequently sold in the U.S. and Japan as part of variable annuity products.
The last decade the literature on pricing and risk management of these options evolved.
Until now, for pricing these options generally a geometric Brownian motion for equity prices is
assumed. However, given the long maturities of the insurance contracts a stochastic volatility
model for equity prices would be more suitable. In this paper closed form expressions are derived
for prices of guaranteed annuity options assuming stochastic volatility for equity prices and either
a 1-factor or 2-factor Gaussian interest rate model. The results indicate that the impact of ignoring
stochastic volatility can be significant.
Keywords: Guaranteed Annuity Option (GAO), Guaranteed Minimum Income Benefit
(GMIB), Stochastic Volatility, Stochastic Interest Rates, Variable Annuities.