| Variable annuities are in the spotlight in today's insurance market. The tax deferral feature and the absence of the investment risk for the insurer (while keeping the possibility of investment benefits) boosted their popularity. They represent the sensible way found by the insurance industry to compete with other stock market and financial intermediaries. A variable annuity is an investment wrapped with a life insurance contract. An insurer who sells variable annuities bears two different types of risk. On one hand, he deals with a financial risk on the investment. On the other hand there exists an actuarial (mortality) risk, given by the lifetime of the insured. Should the insured die, the insurer has to pay a possible claim, depending on the options elected (return of premium, reset, ratchet, roll-up). In the Black-Scholes model, the share price is a continuous function of time. Some rare events (which are rather frequent lately), can accompany jumps in the share price. In this case the market model is incomplete and hence there is no perfect hedging of options. I considered a simple market model with one riskless asset and one risky asset, whose price jumps in different proportions at some random times which correspond to the jump times of a Poisson process. Between the jumps the risky asset follows the Black-Scholes model. The mathematical model consists of a probability space, a Brownian motion and a Poisson process. The jumps are independent and identically distributed. The approach consists of defining a notion of risk and choosing a price and a hedge in order to minimize the risk. In the dual market (insurance and financial) the risk-minimizing strategies defined by Follmer and Sondermann and the work of Moller with equity-linked insurance products are reviewed and used for variable annuities, with death or living benefits.; The theory of incomplete markets is complex and intriguing. There are many interesting connections between such models and game theory, while the newest and maybe the most powerful research tool comes from economics, the utility function (tastes and preferences). |
