We consider the N step binomial tree model of stocks. Call options and put options of European and American type are computed explicitly. With appropriate scaling in time and jumps,  convergence of the stock prices and the option prices are obtained as N-> infinite. The obtained convergence is the Black-Scholes model and, for the particular case of European call option, the Black-Scholes formula is obtained. Furthermore, the Black-Scholes partial differential equation is obtained as a limit from the N step binomial tree model. Pricing of American put option under the Black-Scholes model is obtained as a limit from the N step binomial tree model. With this thesis, option pricing under the Black-Scholes model is achieved not by advanced stochastic analysis but by elementary, easily understandable probability computation. Results which in elementary books on finance are mentioned briefly are here derived in more details. Some important Java codes for N step binomial tree option prices are constructed by the author of the thesis.