Financial derivatives have been widely used for investments or transfer risks among different parties. It has been an important topic to determine their prices.

The celebrated model by Fisher Black, Robert Merton and Myron Scholes is the progenitor to modern finance. However, its application is restricted due to the constant volatility assumption. Therefore, the uncertain volatility models and worst case scenario analysis are introduced [4] and [32] to consider the amount of risk in derivative pricing.

In this dissertation, we discuss the asymptotic behaviors of the worst case scenario price under an uncertain volatility model. We are mainly interested in how the worst case scenario price behaves as the degree of model uncertainty vanishes. We derive the derivative of the worst case scenario price with respect to the interval size: epsilon at epsilon = 0. This study produces an approximation of the worst case scenario price or the worst case scenario volatility process. If the volatility interval is small in the model, we need only to solve two linear PDEs to approximate the solution of a fully nonlinear PDE: Black-Scholes-Barenblatt equation.