We solve an exponential utility maximization problem with unbounded payoffs and portfolio constraints, via the theory of quadratic backward stochastic differential equations with unbounded terminal data. This generalizes the previous work of Hu et al. (2005) [ Ann. Appl. Probab. 15, 1691--1712] from the bounded to an unbounded framework. Furthermore, we study utility indifference valuation of financial derivatives with unbounded payoffs, and derive a novel convex dual representation of the prices. In particular, we obtain new asymptotic behavior as the risk aversion parameter tends to either zero or infinity. This talk is based on the joint work with Ying Hu and Shanjian Tang.