The purpose of this work is to analyze some of the most important econometric models for the option pricing. The common feature of these approaches is the intensive use of conditionally heteroscedastic models of the class GARCH (Generalized Autoregressive Conditional Heteroskedasticity, Bollerslev (1986)) and its extensions. Option pricing is a field which is currently receiving more and more increasing interest. Options’ evaluations are very important for a financial analyst or a risk manager who works in an big financial institutions or in a merchant bank. It is a field which uses a large variety of statistical tools and financial and mathematical models which allow to manage the risk and to make forecasts. One of the most used mathematical model for pricing options is the model of Black and Scholes (1973), in which the price of a call or put option depends by five key variables: the underlying price (S), the exercise price (K), the risk-free interest rate (r), the maturity of the option (T) and the volatility of the underlying (sigma). In this work we will see in particular what happens if we consider a conditionally heteroscedastic structure for the volatility of the underlying asset, estimating several GARCH models and used the forecasts of the volatility of the underlying into the Black-Scholes model. We reported below a list of what we will present in this work:

1. In the first chapter we will introduce ARCH (Auto-Regressive Conditional Heteroskedasticity), GARCH and their extensions. In this brief review we will show the properties of these models, used at the end for pricing options;

2. In the second chapter we will show some properties of these instruments and at the end the put-call relation or also called parity relation; we will examine briefly the mechanics of options’ market. In particular we will examine three markets: Italian, US and UK markets. English market is the object of our analysis;

3. In the third chapter we will show the most famous option pricing model: Black and Scholes model, and its limitations, which are used for talking about the possible alternatives; we will present the (ad hoc) BS model of Dumas, Fleming and Whaley (1998), the BS model with the forecasts of the volatility of several GARCH models (Engle, Kane and Noh (1994)) and at the end we will introduce the «GARCH option pricing» of Duan (1995) and the Heston and Nandi (2000) model;

4. In the fourth chapter we will show the applications to real data. They are FTSE 100 Index options, kindly lended by the center of computational finance «HERMES» of the University of Cyprus. We will evaluate the options using the approach of Engle, Kane and Noh (1994), that is we will generate the forecasts of the volatility (Out-of-Sample and In-Sample) of the underlying asset of various GARCH models (ARCH, GARCH, E-GARCH, GJR-GARCH, T-GARCH, I-GARCH, NA-GARCH, AV-GARCH, AP-ARCH, ALL-GARCH) and we will plug these forecasts into the BS model. We will evaluate the performances of each model with some loss functions. Estimation of the models, statistics and all the graphics are made with the softwares MATLAB 7.6, Eviews 5.0 and R 2.9.2.