This thesis studies binomial and trinomial lattice approximations in Black-Scholes type option pricing models. Also, it covers the basics of these models, derivations of model parameters by several methods under different kinds of distributions. Furthermore, the convergence of binomial model to normal distribution, Geometric Brownian Motion and Black-Scholes model isdiscussed. Finally, the connections and interrelations between discrete random variables under the Lattice approach and continuous random variables under models which follow Geometric Brownian Motion are discussed, compared and contrasted.