Options are financial derivatives representing a contract which gives the right to the holder, but not the obligation, to buy or sell an underlying asset at a pre-defined strike price during a certain period of time. These derivative contracts can derive their value from almost any underlying asset or even another derivative: stock-options, options on bonds, swap options (options on swaps), weather options, real options and many others. Options have existed for a long period of time but they became widely popular after Fisher Black, Myron Scholes and Robert Merton developed a theoretical pricing model in 1973 known as the Black-Scholes model. Options became a standardized product traded on the Chicago board of options Exchange (CBOT) through the clearing house guarantees. Nowadays, options are both market and OTC (over the counter) traded and are mainly used for portfolio hedging and speculation. In this paper I am going to study market risk management from the perspective of options trader, and I will show how to describe the risk characteristics of plain vanilla European stock options contracts by going through the “Greeks” which are defined as quantities that represent option’s sensitivity to risk. Finally, I will construct portfolios that will eliminate these risks.