In contrast to the price of derivative securities in a complete financial market, which can be uniquely determined by constructing a portfolio that hedges the payoff, the price of insurance risk typically is not unique. This is because the price of insurance risk is also determined by supply and demand for insurance products in the market. No mathematical model to date has fully captured the interactions between insurance markets and financial markets that characterize modern corporate practice.
Young and Zariphopoulou (2001) introduced an expected utility approach to price dynamic insurance risk. Their valuation is based on comparing maximal expected utility under two scenarios: with and without incorporating the sale of a given insurance product.
Their work gives reserve prices for the company and for the customer; that is, upper and lower bounds on the price of the insurance product so that the transaction can occur. However, Young and Zariphopoulou did not specify which particular value in the interval between reserve prices should be adopted by the insurer. In this paper, we will solve this problem and generalize their work by incorporating a demand curve which addresses the market demand of a certain insurance product at a given price. In this way, insurance policies are treated en masse and their price can be calculated using the Dynamic Programming Principle.