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In finance, a derivative is a financial instrument whose value is derived from one or more underlying assets. An option is a contract which gives the owner the right, but not the obligation, to buy or sell the asset at a specified strike price at the specified date. The derivative itself is just a contract between two or more parties. Its value is determined by fluctuations in the value of the underlying asset. This price is chosen so that the value of the contract to both sides is zero at the outset, which means that the price is fair, so neither party is taking advantage of the other. Hence, numerical methods are needed for pricing options in cases where analytic solutions are either unavailable or not easily computable. The subject of
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This method is widely used as it is able to handle a variety of conditions. Finite difference methods were first applied to option pricing by Eduardo Schwartz in1977. In general, finite difference methods are used to price options by approximating the differential equation that describes how the option price moves over time by a set of difference equations. This method arises since the option value can be modeled by partial differential equations, such as the Black-Scholes PDE. This approach has the same level of complexity tree methods. The application of Monte Carlo method to option pricing was by Phelim Boyle in 1977. In terms of theory, Monte Carlo valuation relies on risk neutral valuation. The technique is to generate several thousand possible random price paths for the underlying asset and via simulation, and to calculate the average payoff of each path. This approach is particularly useful in the valuation of options with complicated features, which would be difficult to value through straightforward Black-Scholes style or tree model. ( reference [3] Valuation of Options)

Each of these methods has its own advantages and disadvantages. The comparison of accuracy and consistence are presented and suitable method for each situation is discussed.

Then the report briefly goes through some exotic options and implements the numerical solutions with binomial tree method. These options, includes American option which can be exercised any time before the

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