One way to think about what is happening is that we are approximating the terminal distribution of the price of the underlying at the maturity date of the option. At each node of this tree, the value of the option can be found using simple arbitrage arguments, it is possible to construct a portfolio by combining riskless borrowing and the risky asset to duplicate the payoff of an option, which means the price of the option is easy to find.
To value an option using this approach, we specify the number of periods to split the time to maturity into, and then calculate the option using a binomial tree with that number of steps.
and the number of periods , calculate
To find the option price, will ``roll backwards:'' At node , calculate the call price as a function of the two possible outcomes at time . For example, if there is one step,
find the call price at time 0 as