Pricing European Option By Finite Difference Method And Monte-Carlo Method

European options known as most simple vanilla options are most commonly traded in the financial market. They are the most basic type of option, with relatively simple features and payoffs. In fact we can find the closed solution for this kind of options directly, but science this is the most basic type of option, and there are many kind of more complicated options exists in the market which we are not able to develop a formula analytically to calculate the price, for those we need to find some numerical way to estimate the price, and all those pricing model based on the model for European option pricing model. Numerical methods form an important part of options pricing and especially in cases where there is no closed form analytic formula. there are two of the primary numerical methods that are currently used by financial professionals for determining the price of an options namely Monte Carlo method and finite difference method. Then we compare the convergence of the two methods to the analytic Black-Scholes price of European option. Monte Carlo method is good for pricing exotic options while implicit (Crank Nicolson)finite difference method is unconditionally stable,more accurate and converges faster than Monte Carlo method when pricing standard options. So in this paper we foxed on to determine a numerical method for pricing the European-style options whose values satisfy the Black-Scholes pricing model. We mainly introduced finite difference method(explicit and implicit method) and Monte-Carlo methods. We discuss both advantages and disadvantages, we also discussed the variance reduction methods(important sampling, control variance, antithetic variables, while using Monte-Carlo method) to improve the accuracy. Moreover we gave the implementation of option price by both method. At the end we also little introduced the numerical method for pricing Asian option based on the method we developed for European option. This paper give a good starting point to learn more about pricing more complicated options (exotic options) such as Asian option, digital option.