Multi-Period Binomial Model

In this lesson, we learned about the multi-period binomial model.  This model is just like the 1-period binomial model except that instead of only having two time periods it can have n time periods.  Below is an example of a three period binomial model:

This is just a series of 1 period models put together. To get the risk-neutral probabilities, we can just use the same formulas as we did in the last module.


We next priced a European call option based on the risk-neutral probabilities.

If the risk-neutral probability of the stock price moving up is q and the probability of the stock price moving down is (1-q), then the probability of it moving up in each of the periods is q^3, the probability of it moving down in each period is (1-q)^3.  There are three paths the stock could take to end at $107 and three paths the stock could take to end at $93.46, so both of these have a 3 in front of their probabilities (3q^2(1-q) and 3q(1-q)^2, respectively).

In this image above, the payouts for the call option are displayed in blue above the final stock prices. We are going to work backwards from t=3 to price a 1-period call option from each node.  These are also displayed above the nodes at t=0, t=1, and t=2 to compute value of the option at t=0 to be equal to 6.57.

We could also do all the calculations as one total calculation using the following formula:

where 1/R^3 is simply the discount factor.  The full formula would be:

1/R^3 *[[q^3 *22.5] + [3q^2(1-q) * 7] + [3q(1-q)^2 * 0] + [(1-q)^3 *0]]

which equal 6.57 after calculation, as we found earlier.

Remember that this is the arbitrage-free value of the call option that does not depend on the probability  values. We will discuss creating a replicating strategy for a multi-period binomial model in later lessons.