The purpose of this report is to conduct an analysis on various methods of pricing certain financial derivatives, as well as hedging European options. A derivative is a security whose profit depends on the fluctuation in value of one or more underlying assets. Over the last decade or so derivatives have become very important financial instruments for transferring financial risk. Derivatives are generally used to hedge risk exposure or to speculate by taking on additional risk in the hope of exploiting this risk for profit. Derivatives can be traded directly on financial exchanges or over-the-counter with investment banks. It is also common for many financial products to have derivatives built into their design, for example many investment contracts offered by insurance companies offer some sort of guarantee where the initial investment (and possibly future contributions) are guaranteed. This can be seen as a put option on the fund's investment performance. Determining the price/value of these options/guarantees is essential to the success of these products. For example, the demise of Equitable Life (the UK's oldest life insurance company) can be attributed to the poor rating of the guaranteed rates it offered on some annuity products; it subsequently closed its operations in 2000. In this report, we analyze derivatives such as forwards and futures contracts, as well as various options. A forward contract on a stock or commodity is a contract to buy/sell a specific quantity of that stock or commodity for a specific price (known as the delivery price K) at a specified future date (expiry T). Forward contracts are generally used to lock in the price of a commodity and eliminate the uncertainty surrounding… middle of the paper… the specified price. An American option can be exercised at any time before expiration, while a European option can only be exercised at expiration. A European call option gives the buyer the right but not the obligation to buy the underlying at a prescribed price K (the strike price). at expiry time T. A European put option gives the buyer the right but not the obligation to sell the underlying at a prescribed price K at expiry time T. The profit of an option is always non-negative and for European options is determined by the difference between the strike price and the stock price at expiration T. American options have similar advantages, except that we must take into account that American options can be exercised at any time until expiration . In the following sections we will discuss the methods used for pricing options. We will also analyze different ways of hedging European options.