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Valuation of Guaranteed Annuity Options using a Stochastic Volatility Model for Equity Prices

Alexander van Haastrecht, Richard Plat and Antoon Pelsser

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.