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the price of the option is calculated at maturity and recursively at each node up to the initial value, by discounting backwards at the risk free rate and respective probabilities. Due to the characteristic of American options, the model has to check if it is optimal to exercise the option at each node or if it has the advantage to continue to the next one, for example on the case of dividend payments.

In the case that it is optimal to exercise the option at a certain node, its price will be equal to the intrinsic value at that same node. Every node will be checked for the optimality of exercising the option or not, until we have reached the initial point where we want to price the option.

The model starts being built for a American option of a non dividend paying stock and after that the scenario of dividend payments and optimal early exercise strategy is considered.

As referenced before the stock goes up and down by a certain amount form one period to the next, if u is the up movement and d the down movement, then they can be calculated as, (3.3.1.1) and (3.3.1.2) as in Cox, Ross and Rubinstein (1979). In no arbitrage conditions it is possible to calculate the probability of the up and down movements, with the up being defined as, (3.3.1.3) where from the definition of probability and the down movement as (3.3.1.4).The tree formed using these specifications from Cox, Ross and Rubinstein (1979), can have the following graphical representation。

The option is price is calculated from the asset price binomial tree. The maturity boundary condition for an American option, is that the payoff is equal to , we already have S at each maturity node from the asset price model, so we can calculate backwards the price of the option as the expectation of the future payoff of the option.

At each node we calculate the expectation of the future payoffs, where the price of the option will be a compound of expectations. These can be represented by the multi period case for a call as in Cox, Ross and Rubinstein (1979)。

The option prices are calculated as the expectation of the option's future payoffs using their respective weighted risk neutral probabilities of an up movement and a down movement and then discounted at the risk free rate r. The Binomial value is found for each node, starting at the final time step, and working backwards to the。

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