In recent decades, there has been a growing interest for utility indifference based approaches to solve the question of pricing of derivatives in incomplete markets. In this paper we consider a stochastic volatility model defined as a positive non-Gaussian Ornstein-Uhlenbeck process, and price Call and Put options using the indifference methodology in the case of exponential utility. The purpose of the study is to investigate empirically the implied risk aversion for a representative agent in the option market, as a function of time to maturity and strike price. Our studies are based on price data for two companies, Microsoft and Volvo, where we calibrate the stochastic volatility model using historical price returns. The implied risk aversion is found by inverting numerically the indifference pricing equation, given observed option prices. The numerical inversion involves solving an integro-partial differential equation. We find that the option prices in the market are basically set by the issuer, in the sense that it is the issuer's indifference prices that matches the market prices. Since the stochastic volatility model explains the stylized facts of returns rather well, we expect the implied risk aversion to be rather flat with respect to maturity and strike price of the options. We find on the contrary a clear smile effect for short dated options, which may be explained by the issuer's fear of a market crash (in the case of the issuance of a Put option). Although the stochastic volatility model explains the heavy-tails of the returns, the crash risk seems to be unexplained by the stochastic volatility model.