The value of knowing the market price of risk

We study an optimal allocation problem in a financial market with one risk-free and one risky asset, when the market is driven by a stochastic market price of risk. The problem is set in continuous time, for an investor with a constant relative risk aversion utility, under two scenarios: when the market price of risk is observable (the full information case), and when it is not (the partial information case). The corresponding market models are complete in the partial information case and incomplete under full information. We study how the access to more accurate information on the market price of risk affects the optimal strategies and we determine the maximum price that the investor would be willing to pay to receive such information. In particular, we examine two cases of additional information, when an exact observation of the market price of risk is available either at time 0 only (the initial information case), or during the whole investment period (the dynamic information case).