Stochastic Analysis of Insurance Products

We study the properties of several insurance products via the methods of stochastic analysis and stochastic control. This dissertation consists of the following three parts: (1) We find the minimum probability of lifetime ruin of an investor who can invest in a market with a risky and a riskless asset and who can purchase a reversible life annuity. The surrender charge of a life annuity is a proportion of its value. Ruin occurs when the total of the value of the risky and riskless assets and the surrender value of the life annuity reaches zero. We find the optimal investment strategy and optimal annuity purchase and surrender strategies in two situations: (i) the value of the risky and riskless assets is allowed to be negative, with the imputed surrender value of the life annuity keeping the total positive; or (ii) the value of the risky and riskless assets is required to be non-negative. (2) We model a retiree as a utility-maximizing economic agent who can invest in a financial market with a risky and a riskless asset and who can purchase or surrender reversible annuities. We define the wealth of an individual as the total value of her risky and riskless assets, which is required to be non-negative during her lifetime. We solve this incomplete market utility maximization problem via duality arguments and obtain semi-analytical solutions. (3) We develop a theory for pricing pure endowments when hedging with a mortality forward is allowed. The hazard rate associated with the pure endowment and the reference hazard rate for the mortality forward are correlated and are modeled by diffusion processes. We price the pure endowment by assuming that the issuing company hedges its contract with the mortality forward and requires compensation for the unhedgeable part of the mortality risk in the form of a pre-specified instantaneous Sharpe ratio. We identify the properties of the prices of pure endowments and the factors that affect hedging efficiency.