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

We construct the contingent claims models that price participating policies with rate guarantees and default risk. These policies are characterized by the sharing of profits from an investment portfolio between the insurer and the policyholders. A certain reserve distribution mechanism is employed to credit interest at or above certain specified guaranteed rate periodically to the policyholders. Besides the reversionary reserve distribution, terminal bonus is also paid to the policyholders if the terminal surplus is positive. However, the insurer may default at maturity and the policyholders can only receive the residual assets. By neglecting market frictions, mortality risk and surrender option, and under certain assumptions on the interest rate crediting mechanism, we are able to find analytic approximation solution to the pricing model using perturbation techniques. We also develop effective finite difference algorithms for the numerical solution of the contingent claims models. Pricing behaviors of these participating policies with respect to various parameters in the pricing models are examined.

We analyze the nature of the dynamic fund protection which provides an investment fund with a floor level of protection against a reference stock index (or stock price). The dynamic protection feature entitles the investor the right to reset the value of his investment fund to that of the reference stock index. The reset may occur automatically whenever the investment fund value falls below that of the reference stock index, or only allowed at pre-determined time instants. The protected funds may allow a finite number of resets throughout the life of the fund, where the reset times are chosen optimally by the investor. We examine the relation between the finite-reset funds and automatic-reset funds. We also analyze the premium and the associated exercise policy of the embedded withdrawal right in protected funds, where the investor has the right to withdraw the fund prematurely. The impact of proportional fees on the optimal withdrawal policies is also analyzed. The holder should optimally withdraw at a lower critical fund value when the rate of proportional fees increases. Under the assumption that the fund value and index value follow the Geometric Brownian processes, we compute the grant-date and mid-contract valuation of these protected funds. Pricing properties of the protected fund value and the cost to the sponsor are also discussed.