In this paper,it is discussed the Europe option pricing formula,and provides a simple method of deriving Black-Scholes option pricing formula that enable those readers only with calculus knowledge to understand.

It is discussed the Europe option pricing formula, under the assumptions that stocks prices process driven by Poisson jump-diffusion process, and the expected rate μ,volatility σ and risk-less rate are functions of time, we obtain the pricing formula and put-call parity of European option.

In the integral formula of Fourier transforms of option pricing formula,by using residues theorem two integrations were simplified into a single numerical integration which has a faster rate of decay.

Based on the principle of control variables method,this paper adopts CV-CRR method of the American option on the basis of Black-Scholes option pricing formula,and it also makes an empirical analysis to prove that the control variables method can be used to greatly improve the operating speed and valuation precision of the standard binary tree method,thus improve the effciency of valuation.

The derivation of Black-Scholes option pricing formula is very complicated,and it needs some advanced mathematical knowledge such as stochastic process,stochastic differential equation.