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# Binomial Option Pricing Model

The binomial option pricing model proceeds from the assumption that the value of the underlying asset follows an evolution such that in each period it increases by a fixed proportion (the up factor) or decreases by another (the down factor). Using a binomial tree one can project all possible values of the underlying asset at the option's expiration date, and from them, all possible final values for the option. To find the current value of the option, one needs to work backwards through the tree starting with the known final option values. The key is to recognize that it is always possible to create a portfolio made up of a position in the underlying asset combined with a position in the lending market that will have the same next period value as the option. The restricted assumptions about the movements in the value of the underlying asset imply that there is enough information to determine the portfolio weights and thus the value of the replicating portfolio. Under the assumption of no-arbitrage, the replicating portfolio must have the same value as the option.

### DETAILS

The weights for the two components in the replicating portfolio are determined by solving two equations in two unknowns. Take any node in the option value tree one period before expiration. From that node, there are two branches: one representing the final value of the option if the underlying goes up in price in the final period (call it ) and the other, the final value of the option if the underlying goes down in price in the final period (call it ). Let represent the investment in the underlying asset in the replicating portfolio and let represent the number of units of money that are borrowed or lent at the one-period interest rate, . Finally, let be the price of the underlying at our chosen node, which we can deduce from the binomial tree for the price of the underlying. Since the value of the replicating portfolio in the final period is to be equal to the value of the option in the final period, the following two equations must hold, and can be solved to obtain expressions for and :

For a call option, the replicating portfolio will consist of a long position in the underlying, partially financed by borrowing in the money market. For a put option, it consists of a short position in the underlying, combined with lending in the money market. To obtain the value of the option at the chosen node, the expressions for and are substituted into the expression for the value of the replicating portfolio in that period, . The same procedure can be done on all the nodes one-period from expiration. Once all the values of the option one-period from expiration are determined, one can move back through the tree and perform the same operation on all the nodes two periods before expiration, and so on, until the single node representing the current value of the option is reached.

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