Black-Scholes equation is a revolutionary concept in the modern financial theory. Financial instruments such as stocks, commodities and derivatives can be evaluated using this model. Option valuation, a part of derivative products, is extremely important to trade in the stocks. The numerical solutions of Black-Scholes equations are used to simulate these options and are addressed in this dissertation. In particular, the discontinuities in the domain are addressed. The discontinuities in options create oscillations near exercise price in the solution. A grid adaptive finite difference technique is developed to evaluate financial options using Black-Scholes equation. The grid is refined near the exercise price to resolve discontinuities in the option evaluation and a coarse grid is generated otherwise. To cope with these uneven space steps, an innovative numerical finite difference schemes are developed named as adaptive explicit, adaptive fully implicit and adaptive Crank Nicolson techniques. These techniques are used to cure oscillations produced by discontinuities in the digital and butterfly options. These techniques are also used to simulate multi-asset digital and butterfly options. The numerical experiments show that the adaptive finite difference method is much more efficient than the method with uniform spacing. The technique reduces the points drastically which in turn decreases the computational cost and makes the algorithms highly efficient