In this paper we consider the problem of the quantile hedging from the point of view of a better informed agent acting on the market. The additional knowledge of the agent is modelled by a filtration initially enlarged by some random variable. By using equivalent martingale measures introduced in  and  we solve the problem for the complete case, by extending the results obtained in  to the insider context. Finally, we consider the examples with the explicit calculations within the standard Black–Scholes model.
2000 AMS Mathematics Subject Classification: Primary: 60H30; Secondary: 60G44.