Solution of the HJB Equations Involved in Utility-Based Pricing

Abstract

In this paper the connection between the utility pricing methodology and risk sensitive control is explored for stochastic volatility models. It is proved that the utility based price of a European option can be written as the difference of the value functions of two different stochastic optimal control problems. The smoothness of those value functions and gradient estimates are proved, to give a complete solution to these problems. As a consequence of these results, the relation with quadratic BSDEs, as well as the description of a risk neutral measure associated with this pricing approach are formalized.