Option pricing plays an important role in modern finance. This paper investigates the uncertain option pricing problems based on uncertainty theory for Liu's uncertain stock model and Peng's mean-reverting stock model which are two basic and representative uncertain stock models in uncertain finance. The pricing formulas of the European and American options are derived by applying the method to calculate the optimistic value of uncertain returns of options instead of the usual method of expected value in the sense of the weighted average. In the end, some numerical experiments are given to illustrate the effectiveness of the obtained results.

How to choose the optimization criterion of the objective function is an important issue for uncertain optimal control. The Hurwicz criterion is a flexible optimization criterion attempting to find the intermediate area between the extremes posed by the optimistic and pessimistic criteria. Based on uncertainty theory, in this paper, we establish a new uncertain optimal control model with jump by making use of Hurwicz criterion to optimize an uncertain objective function. By applying Bellman's principle of optimality, the principle of optimality for the proposed model is presented and then the equation of optimality is derived. Finally, an example is given to show the the effectiveness of the results obtained.