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(1973) and Merton(1973) pricing methods which are the basis for most of this paper assume that the stock returns follow a Geometric Brownian motions, with the stock prices log normally distributed.

The stock returns can be represented by the following stochastic differential equation,Where St is the asset price at time t, is the assets expected return, is the assets instantaneous volatility and Wt is a Wiener process.

Analytical Approximation by Barone Adesi and Whaley (1987) 。

Barone Adesi and Whaley (1987) developed a method to approximate analytically and easily the price of American options. They considered that the American and European option pricing equation is represented by the partial differential equation (3.2.1) developed by Black and Scholes (1987) and Merton (1987)。

Barone Adesi and Whaley (1987) assumed that if this is true, then the early exercise premium of the American option, which is the price difference between the American and the European call option prices (3.2.2), can be represented by the same partial differential equation .

The above equation after some transformation, shown on Barone Adesi and Whaley (1987) paper, and applying an approximation of a term tending to zero, yields the following quadratic equation。

Where (3.2.5), (3.2.6) and (3.2.7). Equation (3.2.4) 'is a second order ordinary differential equation with two linearly independent solutions of the form . They can be found by substituting (3.2.8) into' equation (3.2.4) Barone Adesi and Whaley (1987),When the American option boundary conditions are applied to the above solution and considering , then must be equal to 0 as when the asset price tends to zero so does the option price, resulting in the following American call option pricing equation, Barone Adesi and Whaley (1987)。

From (3.2.9) we have the value for so the only value missing is . This can be calculated interactively considering another boundary condition of American call options. We know that in early exercise the payoff will never be higher than S - X, so from a critical underlying asset value the option payoff curve must be tangent to the S - X curve, which means that below the critical asset value the pricing equation is represented by (3.2.11), Barone Adesi and Whaley (1987).

The algorithm presented by Barone Adesi and Whaley (1987) for the above pricing problem is presented further in the paper in the section dedicated to the implementation of the American option pricing models.

Cox, Ross and Rubinstein (1979) proposed a model where the underlying asset would go up or down from one time step to the next by a certain proportional amount and with a certain probability until maturity. Due to the up and down characteristic of the asset price model these type of models are characterised by a binomial tree or, in the cases of the existence of a third possible movement, they are characterised by a trinomial tree, therefore named as Binomial or Trinomial models。

The price of the option would be recursively derived from maturity, due to the boundary condition as has been referenced before that the price of the option is only known with certainty at maturity.

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