Nella prima parte del nostro lavoro, il surplus di una compagnia d’assicurazione è modellizzato come un processo di Wiener. Consideriamo un contratto d’assicurazione dinamica di solvibilità. Secondo questo contratto, i pagamenti necessari sono effettutati istantaneamente, in modo che il surplus modificato non divenga mai negativo. Matematicamente, questo corrisponde ad introdurre una barriera riflettente in zero. Otteniamo così un’espressione esplicita per il premio netto di un tale contratto.
In the first part of the paper the surplus of a company is modelled by a Wiener process. We consider a dynamic solvency insurance contract. Under such a contract the necessary payments are made instantaneously so that the modified surplus never falls below zero. This means mathematically that the modified surplus process is obtained from the original surplus process by introduction of a reflecting barrier at zero. Theorem 1 gives an explicit expression for the net single premium of such a contract.
In the second part we consider an investment fund whose unit value is modelled by a geometric Brownian motion. Different forms of dynamic investment fund protection are examined. The basic form is a guarantee which provides instantaneously the necessary payments so that the upgraded fund unit value does not fall below a protected level. Theorem 2 gives an explicit expression for the price of such a guarantee. This result can also be applied to price a guarantee where the protected level is an exponential function of time. Moreover it is shown explicitly how the garantee can be generated by construction of the replicating portfolio. The dynamic investment fund garantee is compared to the corresponding put option and it is observed that for short time intervals the ratio of the prices is about 2. Finally the price of a more exotic protection is discussed, under which the guaranteed unit value at any time is a fixed fraction of the maximal upgraded unit value that has been observed until then. Several numerical and graphical illustrations show how the theoretical results can be implemented in practice.