Mixed dynamic and static risk-minimization with an application to survivor swaps

In the traditional setup, the financial market consists of liquid and dynamically traded financial assets. Here, we extend this setup to include an illiquid asset, which may be traded at fixed, discrete times only. Within this setting of mixed dynamic and static hedging, we adopt the criterion of risk-minimization and minimize the so-called risk process at the fixed trading times for the illiquid asset. The optimal mixed dynamic and static risk-minimizing strategies are compared with the optimal dynamic strategies, and certain correction terms that arise, when trading is restricted to discrete time for the illiquid asset, are identified. We apply the technique for a life insurance company whose liabilities are described by a general insurance payment process. Here, the traditional financial market contains a savings account and a zero coupon bond, which may be traded continuously, and an illiquid mortality derivative, traded at fixed times. We provide numerical illustrations with survivor swaps and compare the minimum obtainable risk with the risk for the optimal dynamic strategies.