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Chapter 3 provides a description of the methods used, their advantages, disadvantages and limitations. Here the required equations will be derived and the solution for the pricing of American options will be provided.

Chapter 4 focus on the algorithms used and their implementation on the MatLab environment, also as the procedures for the development of the GUI for easier user interface.

On Chapter 5 results and their comparison are shown for the different methods used, with the required figures to support the numerical answers.

In the final chapter the dissertation is concluded and a summary of the findings is provided, also as with further work on this subject.

Black and Scholes (1973) and Merton (1973) developed the first analytical closed form solution for the pricing of European type options and certain types of American options, such as American call options on non dividend paying stocks. 'The option pricing model developed by Black and Scholes and extended by Merton gives rise to partial differential equations governing the value of an option' Schwartz (1976).

Black and Scholes (1973) develop their model on the basis of the no arbitrage theory, 'If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks' Black and Scholes (1973).

The Black and Scholes (1973) model valued European options on non dividend paying stocks, and with a number of quite restrictive assumptions, constant and known interest rates, the markets are frictionless with no transaction costs and penalties for short selling. The Black and Scholes (1973) model also assumes that the underlying stocks follow a random walk. Due to all this assumptions the pricing model Black and Scholes (1973) proposed was of easy use, and there is only the need to input the required values on the proposed pricing equation. The model they have proposed does not take into consideration early exercise of the option so it is inaccurate for pricing American Options.

One of the most popular analytical approximation models that starts from the Black and Scholes (1973) model and adjusts it to consider the scenario of early exercise strategies is the work by Baron Adesi and Whaley (1987) which was based on the paper by MacMillan (1986).

Baron Adesi and Whaley (1987) consider that the Black and Scholes (1973) partial differential equation must apply to the early exercise premium as this is just the difference between the American and the European option prices, which are also priced by the same partial differential equation. After some transformation they end with an easily solvable through an interactive process second order differential equation.

When closed form solutions, like the Black and Scholes (1973) valuation model cannot be derived, numerical methods must be developed. These are computational methods where the values for the underlying assets are modelled up to maturity and the price of the options is derived from them. In the case of American options this is a complex process, as the modelled price chang本论文由英语论文网提供整理，提供论文代写，英语论文代写，代写论文，代写英语论文，代写留学生论文，代写英文论文，留学生论文代写相关核心关键词搜索。