The objective of this report is to carry out an analysis of the different pricing methods of certain financial derivatives, as well as the hedging of European options. A derivative is a security whose return depends on the fluctuation in value of one or more underlying assets. Over the past decade, derivatives have become very important financial instruments for financial risk transfers. Derivatives are typically used to hedge risk exposure or to speculate by taking on additional risk in the hope of exploiting that 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 incorporate derivatives into their design. For example, many investment contracts offered by insurance companies offer some sort of guarantee in which the initial investment (and possibly future contributions) are guaranteed. This can be considered a put option on the performance of the fund's investments. Determining the price/value of these options/guarantees is critical to the success of these products. For example, the demise of Equitable Life (the UK's oldest life insurer) can be attributed to mispricing the guaranteed rates it offered on some annuity products – it subsequently closed its doors to new ones. business in 2000. In this report, we will analyze derivatives such as futures and futures, as well as various options. A stock or commodity futures contract is a contract to buy/sell a specific quantity of that stock or commodity at a specific price (called the delivery price K) on a specified future date (maturity T). Futures contracts are typically used to lock in the price of a commodity and eliminate uncertainty surrounding the price of paper. 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 maturity T. A European put option gives the buyer the right but not the obligation to sell the underlying at a prescribed price K at maturity T. The payoff of an option is always non-negative and for European options it is determined by the difference between the strike price and the stock price at maturity T. American options have similar returns, except that it must be taken into account that the American options can be exercised at any time until they expire. In the following sections, we will discuss the methods used for option pricing. We will also analyze different ways to hedge European options.