Valuation of Guaranteed Annuity Options in Affine Term Structure Models

Abstract

We propose three analytic approximation methods for numerical valuation of the guaranteed annuity options in deferred annuity pension policies. The approximation methods include the stochastic duration approach, Edgeworth expansion and analytic approximation in affine diffusions. The payoff structure in the annuity policies is similar to a quanto call option written on a coupon bearing bond. To circumvent the limitations of the onefactor interest rate model, we model the interest rate dynamics by a two-factor affine interest rate term structure model. The numerical accuracy and computational efficiency of these approximation methods are analyzed. We also investigate the value sensitivity of the guaranteed annuity option with respect to different parameters in the pricing model. JEL classification: G13; G23

An Analysis of the Guaranteed Withdrawal Benefits for Life Option

“…In contrast, with other widely marketed VA based options such as the Guaranteed Minimum Accumulation Benefit (GMAB) or the Guaranteed Minimum Income Benefit (GMIB)/ Guaranteed Annuity Option (GAO), the underlying investment is effectively annuitized or invested in fixed income instruments upon maturity. See Chu and Kwok [15], for more details about the GAO products.…”

“…For example, GAO schemes, which are basically equity denominated interest rate options, were launched in the UK in a high interest rate environment, but as they drew closer to maturity interest rates plummeted while the stock markets soared, forcing at least one company to close its product to new buyers (Chu and Kwok, [15]). Also, if the GWB is significantly underpriced or raises the possibility of a debilitating loss for the underwriting company, then the related credit-worthiness issues should make potential clients skeptical of the GWB.…”

“…Biffis [7], Chu and Kwok [15], Boyle and Hardy [10], Ballotta and Haberman [5], Pelsser [27] have analysed the GMIB/GAO type option feature offered in the VAs.…”

“…For the SILN model, we use a 2-factor Vasicek Model for the short-rate 12 , as suggested in Chu and Kwok [15] for pricing Guaranteed Annuity Options (GAO). This short rate model (under the risk-neutral measure) is given by…”

“…This allows us to use time steps as large as one year in the Monte Carlo simulations used to price the GWB and makes the computation much faster. For pricing the GWB, we use the structural-parameter values as specified in Chu and Kwok [15] viz., κ 1 = 0.77, κ 2 = 0.08, σ 1 = 2%, σ 2 = 1%, ρ = −0.7. We also adjust b(t) so as to obtain an intial yield curve to be flat at 3.50% so as to make valuations obtained through this model comparable to the BSM model.…”