Before revision:  The full nonlinear PDE is derived to describe the bid (resp. ask) price dynamic of the claims. Monotone characteristic finite difference schemes for the nonlinear PDE are given,  and the simulations of the bid (resp. ask) prices of contingent claims with uncertainty are implemented.
After revision:  The full nonlinear PDE is derived to describe the bid (resp. ask) price process of the claim. Monotone implicit characteristic finite difference schemes for the nonlinear PDE are given, and the simulations of the bid (resp. ask) prices of contingent claims with uncertainty are implemented.