Exact Solutions and Approximations for Optimal Investment Strategies and Indifference Prices

In this paper a new approach is proposed to determine the optimal strategy for investment in risky assets by a risk averse investor. To generate approximations for such problems in continuous time, we define a sequence of models in discrete time with a finite state space and a restricted class of utility functions for which the exact optimal strategy can be found. We prove that the graphs of optimal policies form a connected subset of two bundles of parallel lines in the plane and that the optimization problem can be reduced to a sequence of simple binary decisions. This allows us to avoid the search over real numbers that is required for every possible value of the state when finite difference schemes for the Hamilton--Jacobi--Bellman equations in continuous time are used. A very efficient calculation scheme is defined which generates the exact solutions for our discrete time approximations, and we use known results from the theory of viscosity solutions to give conditions which guarantee that a sequence of such approximations for a given problem in continuous time converges to the correct limit. We show in a number of examples how the method can be used to find indifference prices in incomplete markets and that our approach can outperform alternative methods that are based on finite difference schemes.