Abstract: In the classical complete finance market, we can provide the unique price for option according to arbitrage-free principle and we can also hedge risk perfectly. With such a hypothesis, we can manage the risk of derivatives efficiently and easily. But in the realistic finance market, many poor risk management cases happen frequently. And the current finance crisis show that the realistic finance market is very complicated and incomplete. In incomplete market, the risk can not be hedged away perfectly, and pricing and hedging problems are intractable. There is no consistent acceptable theory. In this paper, we survey various methods of pricing and hedging in incomplete market for promoting further investigation. We focus on the basic ideas and basic models, and explore the advantages and drawbacks of these methods and the relationship between them. We emphasise the key role that the optimization theory and methods will play and in the meanwhile we also analyse some problems and the gaps in methods that need to make further investigation.

Key words: incomplete market, equilibrium, indifference pricing, partial hedging, good-deal pricing