Quantile hedging for contingent claims in an uncertain financial environment

This paper first studies the quantile hedging problem of contingent claims in an uncertain market model. A special kind of no-arbitrage, that is, the absence of immediate profit, is characterized. Instead of the traditional no-arbitrage targeting the whole market, the absence of immediate profit depends on the confidence level of the portfolio manager for hedging risk. We prove that the condition of absence of immediate profit holds if and only if the initial price of each risky asset lies between the $\alpha$-optimistic value and $\alpha$-pessimistic value of its discounted price at the end of the period. The bounds of the minimal quantile hedging price are derived under the criterion of no-arbitrage in this paper, that is, the absence of immediate profit. Moreover, numerical experiments are implemented to verify that the condition of absence of immediate profit can be a good substitute for the traditional no-arbitrage, since the latter is difficult to achieve. Thus, it may provide a better principle of pricing due to the flexibility from the optional confidence level for the market participants in the increasingly complex financial market.