Mathematical Finance Seminar
November 6, 2003, 5:30 PM to 7:00 PM
Alexander Melnikov, University of Alberta
Hedging methodologies and pricing of equity-linked life insurance contracts
Equity-linked life insurance contracts represent an innovative trend in finance and insurance. These policies combine both financial and insurance risks allowing insurance companies to be more competitive in the modern financial system. Therefore, the pricing of such contracts should be very important for many financial and insurance institutions.
Brennan, Schwartz, and Boyle recognized in their pioneering works the close connection between equity-linked life insurance and option pricing theory created a few years before by Black, Scholes, and Merton. This talk is devoted to how hedging methodologies developed in modern financial mathematics can be exploited to price these mixed contracts.
We study pure endowment life insurance contracts with guarantees. These insurance instruments are based on two assets of the financial market controlled by the Black-Scholes model during the contract period. The first asset is responsible for the maximal size of a future profit, while the second one is more reliable and provides a guarantee for the insured. In this market, the insurance company is considered as a hedger of a maximum of these assets conditioned by the remaining lifetime of the client. We focus primarily on new types of hedging: quantile hedging and efficient hedging with power loss function. These, along with Black-Scholes (fixed guarantee) and Margrabe (flexible guarantee) formulae, create effective actuarial analysis of risks connected with such contracts.
We also show how this approach is extended to a jump-diffusion scheme and discuss some of the connections with the pricing of credit risks and defaultable derivative securities. Finally, we give numerical examples based on financial indices (the Dow Jones Industrial Average and the Russell 2000) to demonstrate how our results can be applied to actuarial practice.