Optimal Investment And Utility Indifference Pricing Based On Stochastic Control Theory

Portfolio selection and asset pricing have always been the core of modern finance theory. As we know, researches on dynamic portfolio investment mainly think about how risk-averse investors choose in different investment environment on the basis of Merton’s problem, and researches on asset pricing in complete market are on the basis of no-arbitrage theory. But when the market is incomplete, there are a number of equivalent martingale measures so we can not find the only price of the asset. A different approach that commonly used in asset pricing problem in incomplete market is utility indifference pricing theory which was introduced by Hodges and Neuberger, and it is this approach that we use in this paper. Therefore, we use stochastic control theory to study the following two questions:First, this paper studies the optimal investment problem with linear consumption. When investors have a linear consumption, they are likely to go bankrupt, but the traditional Merton’s problem did not consider the possibility of bankruptcy.In this paper, we use stochastic control theory to get the optimal investment strategy and the partial differential equation that the value function satisfies, and then we get the display solutions of the optimal investment strategy under exponential utility function and power utility function respectively.Second, this paper use utility indifference pricing theory to study asset pricing in incomplete market. At beginning, we assume that the interest rate is constant, according to the utility indifference pricing theory we get the differential equation that the indifference price of risky asset satisfies, and we can find its display solution under exponential utility function. Further, we assume that the interest rate follows Ho-Lee model, repeat the derivation process we get the partial differential equation that the indifference price of risky asset satisfies in this case.