The utility indifference pricing mechanism is an alternative to traditional no arbitrage valuation methods for derivative contracts in incomplete markets. It yields the price by comparing the maximal expected utilities with and without trading the derivative. In markets with stochastic volatility, the price turns out to be the solution of a certain quasilinear PDE. We study this price using bounds, asymptotic approximations and numerical solutions, and use these to describe its relation to the usual mechanism. The problem is connected to the Merton problem of portfolio optimization with derivative securities, and we describe how the assumption of exponential utility can make it computationally and analytically tractable.